UNIVERSITY OF CALIFORNIA,IRVINE

On the Mathematics of Slow Light

DISSERTATION

submitted in partial satisfaction of the requirementsfor the degree of

DOCTOR OF PHILOSOPHY

in Mathematics

by

Aaron Thomas Welters

Dissertation Committee:Professor Aleksandr Figotin, Chair

Professor Svetlana JitomirskayaProfessor Abel Klein

2011

Chapter 2 c⃝ 2011 Society for Industrial and Applied Mathematics. Reprinted withpermission. All rights reserved.

All other materials c⃝ 2011 Aaron Thomas Welters.

DEDICATION

To Jayme

ii

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS v

CURRICULUM VITAE vi

ABSTRACT OF THE DISSERTATION ix

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Electrodynamics of Lossless One-Dimensional Photonic Crystals . . . . . . . 3

1.2.1 Time-Harmonic Maxwell’s Equations . . . . . . . . . . . . . . . . . . 31.2.2 Lossless 1-D Photonic Crystals . . . . . . . . . . . . . . . . . . . . . 41.2.3 Maxwell’s Equations as Canonical Equations . . . . . . . . . . . . . . 41.2.4 Definition of Slow Light . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Perturbation Analysis of Degenerate Eigenvalues from a Jordan block1 122.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 The Generic Condition and its Consequences . . . . . . . . . . . . . . . . . . 222.3 Explicit Recursive Formulas for Calculating the Perturbed Spectrum . . . . 232.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Spectral Perturbation Theory for Holomorphic Matrix Functions 463.1 On Holomorphic Matrix-Valued Functions of One Variable . . . . . . . . . . 46

3.1.1 Local Spectral Theory of Holomorphic Matrix Functions . . . . . . . 503.2 On Holomorphic Matrix-Valued Functions of Two Variables . . . . . . . . . 633.3 On the Perturbation Theory for Holomorphic Matrix Functions . . . . . . . 68

3.3.1 Eigenvalue Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . 683.3.2 Eigenvector Perturbations . . . . . . . . . . . . . . . . . . . . . . . . 733.3.3 Analytic Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . 79

1The contents of this chapter also appear in [61]. Copyright c⃝ 2011 Society for Industrial and AppliedMathematics. Reprinted with permission. All rights reserved.

iii

4 Canonical Equations: A Model for Studying Slow Light 934.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.1.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.2 Canonical ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.2.2 Energy Flux, Energy Density, and their Averages . . . . . . . . . . . 1024.2.3 On Points of Definite Type for Canonical ODEs . . . . . . . . . . . . 1034.2.4 Perturbation Theory for Canonical ODEs . . . . . . . . . . . . . . . . 105

4.3 Canonical DAEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.3.1 The Correspondence between Canonical DAEs and Canonical ODEs . 108

4.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.5 Auxiliary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Bibliography 196

iv

ACKNOWLEDGMENTS

I would like to begin by thanking my advisor, Professor Aleksandr Figotin. His insight,guidance, and support have been invaluable. The efforts he has put in helping me develop asa mathematician and a researcher went above and beyond the usual duties of an advisor. Thedepth and breadth of his knowledge in mathematics and electromagnetism and his passionfor research is a continual source of inspiration for me. He has profoundly influenced my lifeand my research and for all of this I am eternally grateful.

To the members of my thesis and advancement committees, Professor Svetlana Jitomirskaya,Professor Abel Klein, Professor Ozdal Boyraz, and Dr. Ilya Vitebskiy, I would like to thankthem for their time and thoughtful input.

Thank you also to Adam Larios, Jeff Matayoshi, and Mike Rael, for many useful and enjoy-able conversations.

A special thanks to Donna McConnell, Siran Kousherian, Tricia Le, Radmila Milosavljevic,and Casey Sakasegawa, and the other staff at the UCI math department, for handling themassive amount of paperwork and administrative details that allow me the time to concen-trate on research. Their hard work behind the scenes helps make the UCI MathematicsDepartment the excellent place that it is.

I would also like to thank my family. My parents, Tom and Kathy Welters, and my brothers,Matt and Andy Welters, for being great role models on how to work hard and achievewhile still balancing ones life with respect to family, fun, and career and for instilled inme the importance and joys of family. Thank you so much for all the love, support, andencouragement over the years. I would like to thank my son, Ashton Welters, for making mylife happier than I thought possible. I would especially like to thank my wife and best friend,Jayme Welters. I could not have made it without her encouragement, support, commitment,friendship, love, and laughter. Thank you all that you have done for me over the years, it isappreciated more than I can possibly say in words.

The work on the explicit recursive formulas in the spectral perturbation analysis of a Jordanblock was first published in SIAM Journal of Matrix Analysis and Applications, Volume 32,no. 1, (2011) 1–22 and permission to use the copyrighted material in this thesis has beengranted.

I am thankful to the anonymous referees for helpful suggestions and insightful comments onparts of this research. I am thankful for the University of California, Irvine, where this workwas completed.

This work was supported in part by the Air Force Office of Scientific Research (AFOSR)under the grant FA9550-08-1-0103.

v

CURRICULUM VITAE

Aaron Thomas Welters

PARTICULARS

EDUCATION

University of California Irvine Irvine, CAPh. D. in Mathematics June 2011Advisor: Dr. Aleksandr Figotin

St. Cloud State University St. Cloud, MNB. A. in Mathematics May 2004Magna Cum LaudeAdvisor: Dr. J. -P. Jeff Chen

RESEARCH INTERESTS

My research interests are in mathematical physics, material science, electromagnetics, wavepropagation in periodic media (e.g., photonic crystals), spectral and scattering theory. I spe-cialize in the spectral theory of periodic differential operators, nonlinear eigenvalue problems,perturbation theory for non-self-adjoint matrices and operators, linear differential-algebraicequations (DAEs), block operator matrices, boundary value problems, and meromorphicFredholm-valued operators. I apply the mathematical methods from these areas to studyspectral and scattering problems of electromagnetic waves in complex and periodic struc-tures.

ACADEMIC HONORS

∙ Von Neumann Award for Outstanding Performance as a Graduate Student, UC Irvine, 2010

∙ Mathematics Department Scholarship, SCSU, 2003

PUBLICATIONS

2. A. Welters, Perturbation Analysis of Slow Waves for Periodic Differential-Algebraic of Defi-nite Type. (in preparation)

1. A. Welters, On Explicit Recursive Formulas in the Spectral Perturbation Analysis of a JordanBlock, SIAM J. Matrix Anal. Appl., 32.1 (2011), pp. 1–22.

vi

TALKS

9. UCLA PDE Seminar, University of California, UCLA, CA, February 2011.

8. 2011 AMS/MAA Joint Mathematics Meetings, New Orleans, LA, January 2011.

7. MGSC (Mathematics Graduate Student Colloquium), University of California, Irvine, CA,November 2010.

6. UCI Mathematical Physics Seminar, University of California, Irvine, CA, November 2010.

5. MAA Southern California-Nevada Section Fall 2010 Meeting, University of California, Irvine,CA, October 2010.

4. Arizona School of Analysis with Applications, University of Arizona, Tucson, AZ, March2010.

3. 2010 AMS/MAA Joint Mathematics Meetings, San Francisco, CA, January 2010.

2. UCI Mathematical Physics Seminar, University of California, Irvine, CA, November 2009.

1. SCSU Mathematics Colloquium, St. Cloud State University, St. Cloud, MN, November 2009.

RESEARCH EXPERIENCE

∙ Graduate Student Researcher (GSR), UCI, September, 2007 - April, 2011. The GSRwas supported by the AFOSR grant FA9550-08-1-0103 entitled: High-Q Photonic-CrystalCavities for Light Amplification.

TEACHING EXPERIENCE

TEACHING ASSISTANT

∙ Math2D: Multivariable Calculus. University of California, Irvine, Summer 2006

∙ Math184: History of Mathematics. University of California, Irvine, Spring 2006

∙ Math146: Fourier Analysis. University of California, Irvine, Spring 2006

∙ Math2B: Single Variable Calculus. University of California, Irvine, Fall 2005

∙ Math2B: Single Variable Calculus. University of California, Irvine, Summer 2005

∙ Math2A: Single Variable Calculus. University of California, Irvine, Summer 2005

∙ Math2J: Infinite Series and Linear Algebra. University of California, Irvine, Spring2005

∙ Math2B: Single Variable Calculus. University of California, Irvine, Winter 2005

∙ Math2A: Single Variable Calculus. University of California, Irvine, Fall 2004

vii

INSTITUTIONAL SERVICE

∙ Co-Organizer. “Math Graduate Student Colloquium,” University of California, Irvine;2010-2011

∙ Graduate Recruitment Speaker. “Mathematics Graduate Recruitment Day Event,”University of California, Irvine; April 2010.

COMPUTATIONAL SKILLS

Programming: C/C++, MATLAB, and Maple.

Markup Languages: LaTeX and Beamer.

Operating Systems: Linux/Unix based systems and Windows.

viii

ABSTRACT OF THE DISSERTATION

On the Mathematics of Slow Light

By

Aaron Thomas Welters

Doctor of Philosophy in Mathematics

University of California, Irvine, 2011

Professor Aleksandr Figotin, Chair

This thesis develops a mathematical framework based on spectral perturbation theory for

the analysis of slow light and the slow wave regime for lossless one-dimensional photonic

crystals.

Electrodynamics of lossless one-dimensional photonic crystals incorporating general bian-

isotropic and dispersive materials are considered. The time-harmonic Maxwell’s equations

governing the electromagnetic wave propagation through such periodic structures is shown

to reduce to a canonical system of period differential-algebraic equations (DAEs) depending

holomorphically on frequency which we call Maxwell’s DAEs. In this context, a definition

of slow light is given.

We give a detailed perturbation analysis for degenerate eigenvalues of non-self-adjoint matri-

ces. A generic condition is considered and its consequences are studied. We prove the generic

condition implies the degenerate eigenvalue of the unperturbed matrix under consideration

has a single Jordan block in its Jordan normal form corresponding to that eigenvalue. We

find explicit recursive formulas to calculate the perturbation expansions of the splitting eigen-

values and their eigenvectors. The coefficients up to the second order for these expansions

are given.

ix

An exposition on the spectral theory and spectral perturbation theory for holomorphic matrix

functions is given. This serves as a mathematical background for the tools and theory used

in this thesis.

We give a model for studying slow light based on canonical equations which are any canonical

system of periodic differential (canonical ODEs) or differential-algebraic (canonical DAEs)

equations depending holomorphically on a spectral parameter referred to as the frequency.

For canonical ODEs, we prove formulas connecting the dispersion relation, energy flux, and

energy density in our model to the monodromy matrix of the canonical ODEs. We prove one

of the main results of this thesis relating the existence of (non-Bloch) Floquet solutions of the

canonical ODEs and the occurrence of degenerate eigenvalues with nondiagonalizable Jordan

normal form of the monodromy matrix to the band structure of the dispersion relation near

spectral stationary points. For canonical DAEs, of which it is shown Maxwell’s DAEs are

an example, a theorem is given which shows that the model for canonical DAEs, including

the Floquet, spectral, and perturbation theory, is reduced to the model for canonical ODEs.

x

Chapter 1

Introduction

1.1 Motivation

Technology has progressed to the point at which we can now fabricate materials with di-

mensions on the nanoscale. To give you a perspective on how small this is consider that

one nanometer (nm) is equal to one billionth of a meter (m), the width of a human hair is

about 105 nm, and the wavelength of visible light lies in the approximate range of 400–700

nm. As a result, there has emerged a new class of materials (such as metamaterials [11, 54]),

which allow the possibility of novel electromagnetic properties and effects. For example,

bianisotropic crystals can be used to observe optical singularities [6], photonic crystals [27]

can be used to slow down light [16], negative index materials can be used to create a perfect

lens [50], and metametarials derived from transformation optics can be used to induce elec-

tromagnetic invisibility [21, 22, 40, 51]. The novel electromagnetic properties and effects that

metamaterials such as photonic crystals have, is a fascinating and productive area of research

for not only physicists but for mathematicians as well (see for example [32]). In particular,

research on slow light has received considerable attention recently. With its numerous appli-

1

cations such as solar cell efficiency enhancement [10] or antenna miniaturization [49, 59, 62],

there is a growing need for applied and theoretical studies on slow light phenomena.

Although there are several methods for slowing down light including electromagnetically in-

duced transparency (EIT) and photonic crystals, it is the latter that is becoming of particular

interest to mathematicians. A major reason for this is the mathematics of photonic crystals

has been developed significantly over the last 20 years and this gives us a rigorous mathe-

matical setting in which to begin studying problems involving slow light. Moreover, many

of the results on slow wave phenomenon in photonic crystals are expected to have analo-

gies with wave propagation in general linear dispersive media. Thus in order to facilitate

further applied and theoretical studies on slow light, to better understand slow light from a

mathematical perspective, and to explore analogies with slow waves in periodic media, this

thesis considers a general nondissipative but dispersive model for wave propagation using

an important class of periodic differential and differential-algebraic equations (DAEs) called

canonical equations [31]. As we will show in the next section, this model is general enough

to include electromagnetic waves governed by the time-harmonic Maxwell’s equations for

lossless one-dimensional photonic crystals whose constituent layers can be any combination

of isotropic, anisotropic, or bianisotropic materials with or without material dispersion (i.e.,

frequency-dependent response of materials). This makes our work particularly significant

in the study of slow light since metamaterials are widening the range of potential photonic

crystals that can be fabricated and so a model like ours that has the ability to analysis slow

light phenomena for a broad range of photonic crystals is in need.

2

1.2 Electrodynamics of Lossless One-Dimensional Pho-

tonic Crystals

1.2.1 Time-Harmonic Maxwell’s Equations

Electromagnetic waves will be governed by the time-harmonic Maxwell’s equations (e−i!t

convention, ! ∕= 0) without sources in Gaussian units and Cartesian coordinates (with

respect to the standard orthonormal basis vectors e1, e2, e3 of ℝ3). These equations may be

written in the 2× 2 block operator matrix form (see [5])

⎡⎢⎣ 0 ∇×

−∇× 0

⎤⎥⎦⎡⎢⎣ E(r)

H(r)

⎤⎥⎦ = −i!c

⎡⎢⎣ D(r)

B(r)

⎤⎥⎦ (1.1)

where c is the speed of light in a vacuum, r := (x1, x2, x3) are the spatial variables and

the electric field E, magnetic field H, electric induction D, and magnetic induction B take

values in ℂ3. Here ∇× denotes the curl operator on these fields and it is given by the matrix

operator

∇× :=

⎡⎢⎢⎢⎢⎣0 − ∂

∂x3∂∂x2

∂∂x3

0 − ∂∂x1

− ∂∂x2

∂∂x1

0

⎤⎥⎥⎥⎥⎦ . (1.2)

The linear constitutive relations in 2× 2 block matrix form are⎡⎢⎣ D(r)

B(r)

⎤⎥⎦ = C(x3, !)

⎡⎢⎣ E(r)

H(r)

⎤⎥⎦ , C(x3, !) =

⎡⎢⎣ "(x3, !) �(x3, !)

�(x3, !) �(x3, !)

⎤⎥⎦ , (1.3)

where ", �, �, � are 3× 3 matrix-valued functions representing the electric permittivity, mag-

netic permeability, and magnetoelectric coupling tensors, respectively. Here we have tacitly

3

assumed materials which are plane parallel layers normal to the x3-axis with each layer

consisting of a homogeneous material whose frequency dependency is implicitly indicated.

1.2.2 Lossless 1-D Photonic Crystals

We are considering lossless one-dimensional photonic crystals with dispersive materials and

hence there exists a d > 0 and an open connected set Ω ⊆ ℂ, the frequency domain, with

Ωℝ := Ω ∩ ℝ ∕= ∅ such that

(i) C(x3, !) = C(x3 + d, !), for every ! ∈ Ω and a.e. x3.

(ii) C(x3, !)∗ = C(x3, !), for every ! ∈ Ωℝ and a.e. x3.

(iii) C(⋅, !) ∈M6(L2(T)), C ∈ O(Ω,M6(L2(T)))1.

1.2.3 Maxwell’s Equations as Canonical Equations

As in [16], we seek field solutions of the form

⎡⎢⎣ E(r)

H(r)

⎤⎥⎦ = eik⊥⋅r⊥

⎡⎢⎣ E(x3)

H(x3)

⎤⎥⎦ , (1.4)

where r⊥ = (x1, x2, 0), k⊥ = (k1, k2, 0), and k1, k2 ∈ ℝ. Hence solutions to the time-harmonic

Maxwell’s equations in (1.1) with the field representations (1.4) are solutions to the canonical

equations, which we call Maxwell’s DAEs,

G y′(x3) = V (x3, !)y(x3), (1.5)

1See section 4.5 for notation.

4

where y(x3) is a 6× 1 column vector, G is a 6× 6 singular matrix, and V , the Hamiltonian,

is a 6× 6 matrix-valued function having the block representations

y(x3) =

⎡⎢⎣ E(x3)

H(x3)

⎤⎥⎦ , G = i

⎡⎢⎣ e3×

−e3×

⎤⎥⎦ , V (x3, !) =!

cC(x3, !) +

⎡⎢⎣ k⊥×

−k⊥×

⎤⎥⎦ .(1.6)

Here e3× and k⊥× are the matrices

e3× :=

⎡⎢⎢⎢⎢⎣0 −1 0

1 0 0

0 0 0

⎤⎥⎥⎥⎥⎦ , k⊥× :=

⎡⎢⎢⎢⎢⎣0 0 k2

0 0 −k1

−k2 k1 0

⎤⎥⎥⎥⎥⎦ . (1.7)

In particular, they are a canonical system of differential-algebraic equations (DAEs) with

periodic coefficients that depend holomorphically on the frequency ! where the leading

matrix coefficient G ∈M6(ℂ) and the matrix-valued function V : ℝ× Ω→M6(ℂ) have the

properties:

(i) det(G ) = 0, G ∗ = −G

(ii) V (x3, !)∗ = V (x3, !), for each ! ∈ Ωℝ and a.e. x3 ∈ ℝ

(iii) V (x3 + d, !) = V (x3, !), for each ! ∈ Ω and a.e. x3 ∈ ℝ

(iv) V ∈ O(Ω,M6(L2(T))) as a function of frequency.

In Chapter 4 we will consider these types of equations further.

5

1.2.4 Definition of Slow Light

In this context, Bloch solutions of Maxwell’s DAEs, i.e. solutions of the canonical equations

in (1.5) satisfying

y(x3 + d) = eikdy(x3), (1.8)

for some k ∈ ℂ, give rise to the (axial) dispersion relation

! = !(k). (1.9)

For points (k0, !0) ∈ ℝ2 on the graph of this dispersion relation, i.e., the Bloch variety ℬ, a

Bloch solution y with wavenumber-frequency pair (k0, !0) is said to be propagating with its

group velocity

d!

dk

∣∣∣(k0,!0)

. (1.10)

Under certain reservations [7, 8, 27, 63], its group velocity equals its energy velocity, i.e., the

ratio of its averaged energy flux to its energy density,

d!

dk

∣∣∣(k0,!0)

=1d

∫ d0⟨iG y(x3), y(x3)⟩dx3

1d

∫ d0⟨V!(x3, !0)y(x3), y(x3)⟩dx3

, (1.11)

with its energy flux and its energy density are (up to multiplication by a constant) the

functions of the space variable x3 given by

S = ⟨iG y, y⟩, (1.12)

U = ⟨V!(⋅, !0)y, y⟩, (1.13)

6

respectively, where V! denotes the derivative of the Hamiltonian V with respect to frequency

in the MN(L2(T)) norm and ⟨ , ⟩ is the standard inner product for ℂ6.

We now give an intuitive definition of slow light for lossless one-dimensional photonic crystals.

In Chapter 4 we give a more precise definition using canonical equations. We will treat the

(axial) dispersion relation ! = !(k) as a multi-valued function of the wavenumber k. We

will use the notation d!dk∣(k0,!0)

= 0 to mean (k0, !0) is a stationary point on the graph of the

dispersion relation ! = !(k).

Definition 1 (Slow Light) If (k0, !0) ∈ ℝ2 is a stationary point on the graph of the dis-

persion relation ! = !(k), i.e.,

d!

dk

∣∣∣(k0,!0)

= 0, (1.14)

then any propagating Bloch solution of Maxwell’s DAEs (1.5) with wavenumber-frequency

pair (k, !) satisfying 0 < ∣∣(k, !) − (k0, !0)∣∣ ≪ 1 with its group velocity d!dk∣(k,!)

satisfying

∣d!dk∣(k,!)∣ ≪ c is called a slow wave or slow light.

If (k0, !0) ∈ ℝ2 is a stationary point on the graph of the dispersion relation ! = !(k) then

an open ball B((k0, !0), r) in ℂ2 with 0 < r ≪ 1 is called the slow wave regime.

1.3 Main Results

The objective of this dissertation is to begin developing a mathematical framework based

on spectral perturbation theory for the analysis of slow light and the slow wave regime for

lossless one-dimensional photonic crystals.

In the following we describe the main contributions of this thesis which are contained in

7

Chapters 2 and 4.

Perturbation Analysis of Degenerate Eigenvalues from a Jordan

block

A fundamental problem in the perturbation theory for non-self-adjoint matrices with a degen-

erate spectrum is the determination of the perturbed eigenvalues and eigenvectors. Formulas

for the higher order terms of these perturbation expansions are often needed in problems

which require an accurate asymptotic analysis.

For example, my advisor A. Figotin and his colleague, I. Vitebskiy, considered scattering

problems involving slow light in one-dimensional semi-infinite photonic crystals [3, 12–16].

They found that only in the case of the frozen mode regime could incident light enter a

photonic crystal with little reflection and be converted into a slow wave. This frozen mode

regime was found to correspond to a stationary inflection point of the dispersion relation

and a 3 × 3 Jordan block in the Jordan normal form of the unit cell transfer matrix – the

monodromy matrix of the reduced Maxwell’s equations given in [16, §5] or [16, p. 332, (180)]

which are canonical ODEs although not in canonical form. In this setting, the eigenpairs of

the monodromy matrix corresponded to Bloch waves and their Floquet multipliers. Thus in

order for them to rigorously prove the physical results and provide a better understanding of

the very essence of the frozen mode regime, they needed an asymptotic analytic description

of the frozen mode regime which required a sophisticated mathematical framework based on

the spectral perturbation theory of a Jordan block. Unfortunately, at the time when [16] was

written such a theory did not exist and hence this was a big motivating factor for Chapter

22 of this thesis.

2The contents of this chapter also appear in [61]. Copyright c⃝ 2011 Society for Industrial and AppliedMathematics. Reprinted with permission. All rights reserved.

8

In Chapter 2 of this thesis we to develop the spectral perturbation theory of a Jordan block

and address the following:

1. Determine a generic condition that allows a constructive spectral perturbation analysis

for non-self-adjoint matrices with degenerate eigenvalues.

2. What connection is there between that condition and the local Jordan normal form of

the matrix perturbation corresponding to a perturbed eigenvalue?

3. Does there exist explicit recursive formulas to determine the perturbed eigenvalues and

eigenvectors for non-selfadjoint perturbations of matrices with degenerate eigenvalues?

The statement of my main results regarding those three issues are contained in Theorem 1

and Theorem 2 of this thesis which are Theorem 2.1 and Theorem 3.1 from [61]. In partic-

ular, I have developed a constructive perturbation theory for non-self-adjoint matrices with

degenerate eigenvalues and found explicit recursive formulas to calculate the perturbation

expansions of the splitting eigenvalues and their eigenvectors, under a generic condition.

Canonical Equations: A Model for Studying Slow Light

We established above that the study of slow light, i.e., slow waves for lossless one-dimensional

photonic crystals is reduced to the study of Maxwell’s DAEs (1.5) near stationary points

of the dispersion relation, i.e., in the slow wave regime. As Maxwell’s DAEs are canonical

equations, it will be beneficial to use canonical equations to study slow wave propagation. By

analogy with that physical model, we get a general mathematical model for wave propagation

in periodic structures in which we can study the dispersion relation, band structure, spectral

stationary points, slow waves, and the slow wave regime. This is exactly what we do in

Chapter 4. The contents of this chapter is part of the paper [60] currently in preparation.

9

One of the major contribution in Chapter 4 of this thesis is Theorem 48. Combined with

Theorem 35, Theorem 47, and Theorem 50, it is our answer to the question:

1. How are the analytic properties of the dispersion relation and the degree of flatness of

spectral bands near stationary points related to (non-Bloch) Floquet solutions and the

occurrence of degenerate eigenvalues and a nondiagonalizable Jordan normal form for

the monodromy matrix of the canonical equations.

To answer this question and to proof Theorem 48 we need some deep results in the spectral

perturbation theory for holomorphic matrix functions. We give an exposition of this theory

in Chapter 3.

1.4 Overview

This thesis is organized in the following manner.

Chapter 2 concerns the perturbation analysis of non-self-adjoint matrices with degenerate

eigenvalues. A generic condition is considered and its consequences are studied. It is shown

that the generic condition implies the degenerate eigenvalue of the unperturbed matrix under

consideration has a single Jordan block in its Jordan normal form corresponding to the

degenerate eigenvalue. Explicit recursive formulas are given to calculate the perturbation

expansions of the splitting eigenvalues and their eigenvectors. The coefficients up to the

second order for these expansions are conveniently listed for quick reference.

Chapter 3 is an exposition on the spectral theory and spectral perturbation theory for holo-

morphic matrix functions. Its content is the required background material needed in Chapter

4 to prove the main result for that chapter, namely, Theorem 48.

Chapter 4 formulates and studies a model for slow wave propagation in period structures

10

using canonical equations. Both canonical ODEs and canonical DAEs are considered. Elec-

tromagnetic wave propagation in lossless one dimensional photonic crystals was discussed in

section 1.2 as a example for the need to include DAEs into the model. Motivation for the

definitions in the model was given by a consideration of their relevance in the example.

Next, in section 4.2, canonical ODEs are considered. Results are given on energy flux and

energy density. Then in section 4.2.3 an important theorem, Theorem 47, is given on points

of definite type for canonical ODEs. In particular, the result justifies the use of the term

definite type and states that to each point of definite type of the canonical ODEs there exists

a neighborhood of such points. This is followed up immediately in the next section, section

4.2.4, with the main result of the chapter, Theorem 48, pertaining to the perturbation theory

for canonical ODEs near points of definite type. As a corollary we give a result that connects

the generic condition of Chapter 2 to points of definite type and the Jordan normal form of

the monodromy matrix of the canonical ODEs.

Next, in section 4.3, canonical DAEs are considered. A theorem is stated that tells us the

theory of canonical DAEs is reduced to the study of canonical ODEs including the Floquet,

spectral, and perturbation theory.

Finally, section 4.4 gives the proofs of all the statements in this chapter. The use of the

spectral perturbation theory of holomorphic matrix functions is use to prove the main result

of the chapter, Theorem 48.

All of the work done here was completed under the guidance and supervision of Professor

Aleksandr Figotin.

11

Chapter 2

Perturbation Analysis of Degenerate

Eigenvalues from a Jordan block1

2.1 Introduction

Consider an analytic square matrix A (") and its unperturbed matrix A (0) with a degenerate

eigenvalue �0. A fundamental problem in the analytic perturbation theory of non-selfadjoint

matrices is the determination of the perturbed eigenvalues near �0 along with their cor-

responding eigenvectors of the matrix A (") near " = 0. More specifically, let A (") be a

matrix-valued function having a range in Mn(ℂ), the set of n × n matrices with complex

entries, such that its matrix elements are analytic functions of " in a neighborhood of the

origin. Let �0 be an eigenvalue of the matrix A (0) with algebraic multiplicity m ≥ 1. Then

in this situation, it is well known [4, §6.1.7], [28, §II.1.8] that for sufficiently small " all the

perturbed eigenvalues near �0, called the �0-group, and their corresponding eigenvectors may

be represented as a collection of convergent Puiseux series, i.e., convergent Taylor series in

1The contents of this chapter also appear in [61]. Copyright c⃝ 2011 Society for Industrial and AppliedMathematics. Reprinted with permission. All rights reserved.

12

a fractional power of ". What is not well known, however, is how we compute these Puiseux

series when A (") is a non-selfadjoint analytic perturbation and �0 is a defective eigenvalue

of A (0). There are sources on the subject like [4, §7.4], [41], [47], [57, §32], and [58] but it

was found that there lacked explicit formulas, recursive or otherwise, to compute the series

coefficients beyond the first order terms. Thus the fundamental problem that this paper

addresses is to find explicit recursive formulas to determine the Puiseux series coefficients

for the �0-group and their eigenvectors.

This problem is of applied and theoretic importance, for example, in studying the spectral

properties of dispersive media such as photonic crystals. In particular, this is especially true

in the study of slow light [16, 49, 62], where the characteristic equation, det (�I − A (")) = 0,

represents implicitly the dispersion relation for Bloch waves in the periodic crystal. In this

setting " represents a small change in frequency, A(") is the Transfer matrix of a unit cell, and

its eigenpairs, (�("), x(")), correspond to the Bloch waves. From a practical and theoretical

point of view, condition (2.1) on the dispersion relation or its equivalent formulation in

Theorem 1.(i) of this paper regarding the group velocity for this setting, arises naturally in

the study of slow light where the Jordan normal form of the unperturbed Transfer matrix,

A(0), and the perturbation expansions of the eigenpairs of the Transfer matrix play a central

role in the analysis of slow light waves.

Main Results

In this paper under the generic condition,

∂

∂"det (�I − A ("))

∣∣(",�)=(0,�0)

∕= 0, (2.1)

we show that �0 is a non-derogatory eigenvalue of A(0) and the fundamental problem men-

tioned above can be solved. In particular, we prove Theorem 1 and Theorem 2 which together

13

state that when condition (2.1) is true then the Jordan normal form of A (0) corresponding

to the eigenvalue �0 consists of a single m×m Jordan block, the �0-group and their corre-

sponding eigenvectors can each be represented by a single convergent Puiseux series whose

branches are given by

�ℎ (") = �0 +∞∑k=1

�k

(�ℎ"

1m

)kxℎ (") = �0 +

∞∑k=1

�k

(�ℎ"

1m

)k

for ℎ = 0, . . . ,m − 1 and any fixed branch of "1m , where � = e

2�mi, {�k}∞k=1 ⊆ ℂ, {�k}∞k=0 ⊆

ℂn, �1 ∕= 0, and �0 is an eigenvector of A (0) corresponding to the eigenvalue �0. More

importantly though, Theorem 2 gives explicit recursive formulas that allows us to determine

the Puiseux series coefficients, {�k}∞k=1 and {�k}∞k=0, from just the derivatives of A (") at

" = 0. Using these recursive formulas, we compute the leading Puiseux series coefficients up

to the second order and list them in Corollary 4.

The key to all of our results is the study of the characteristic equation for the analytic matrix

A (") under the generic condition (2.1). By an application of the implicit function theorem,

we are able to derive the functional relation between the eigenvalues and the perturbation

parameter. This leads to the implication that the Jordan normal form of the unperturbed

matrix A (0) corresponding to the eigenvalue �0 is a single m×m Jordan block. From this,

we are able to use the method of undetermined coefficients along with a careful combinatorial

analysis to get explicit recursive formulas for determining the Puiseux series coefficients.

We want to take a moment here to show how the results of this paper can be used to

determine the Puiseux series coefficients up to the second order for the case in which the

non-derogatory eigenvalue �0 has algebraic multiplicity m ≥ 2. We start by putting A (0)

14

into the Jordan normal form [35, §6.5: The Jordan Theorem]

U−1A (0)U =

⎡⎢⎣ Jm (�0)

W0

⎤⎥⎦ , (2.2)

where (see notations at end of §1) Jm (�0) is an m ×m Jordan block corresponding to the

eigenvalue �0 and W0 is the Jordan normal form for the rest of the spectrum. Next, define

the vectors u1,. . . , um, as the first m columns of the matrix U ,

ui := Uei, 1 ≤ i ≤ m (2.3)

(These vectors have the properties that u1 is an eigenvector of A (0) corresponding to the

eigenvalue �0, they form a Jordan chain with generator um, and are a basis for the alge-

braic eigenspace of A (0) corresponding to the eigenvalue �0). We then partition the matrix

U−1A′(0)U conformally to the blocks Jm (�0) and W0 of the matrix U−1A (0)U as such

U−1A′(0)U =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∗ ∗ ∗ ⋅ ⋅ ⋅ ∗ ∗ ⋅ ⋅ ⋅ ∗...

......

. . ....

.... . .

...

∗ ∗ ∗ ⋅ ⋅ ⋅ ∗ ∗ ⋅ ⋅ ⋅ ∗

am−1,1 ∗ ∗ ⋅ ⋅ ⋅ ∗ ∗ ⋅ ⋅ ⋅ ∗

am,1 am,2 ∗ ⋅ ⋅ ⋅ ∗ ∗ ⋅ ⋅ ⋅ ∗

∗ ∗ ∗ ⋅ ⋅ ⋅ ∗ ∗ ⋅ ⋅ ⋅ ∗...

......

. . ....

.... . .

...

∗ ∗ ∗ ⋅ ⋅ ⋅ ∗ ∗ ⋅ ⋅ ⋅ ∗

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (2.4)

Now, by Theorem 1 and Theorem 2, it follows that

am,1 = −∂∂"

det (�I − A (")) ∣(",�)=(0,�0)( ∂m

∂�mdet(�I−A("))∣(",�)=(0,�0)

m!

) . (2.5)

15

And hence the generic condition is true if and only if am,1 ∕= 0. This gives us an alternative

method to determine whether the generic condition (2.1) is true or not.

Lets now assume that am,1 ∕= 0 and hence that the generic condition is true. Define f (", �) :=

det (�I − A (")). Then by Theorem 2 and Corollary 4 there is exactly one convergent Puiseux

series for the perturbed eigenvalues near �0 and one for their corresponding eigenvectors

whose branches are given by

�ℎ (") = �0 + �1

(�ℎ"

1m

)+ �2

(�ℎ"

1m

)2

+∞∑k=3

�k

(�ℎ"

1m

)k(2.6)

xℎ (") = x0 + �1

(�ℎ"

1m

)+ �2

(�ℎ"

1m

)2

+∞∑k=3

�k

(�ℎ"

1m

)k(2.7)

for ℎ = 0, . . . ,m − 1 and any fixed branch of "1m , where � = e

2�mi. Furthermore, the series

coefficients up to second order may be given by

�1 = a1/mm,1 =

(−

∂f∂"

(0, �0)1m!

∂mf∂�m

(0, �0)

)1/m

∕= 0, (2.8)

�2 =am−1,1 + am,2

m�m−21

=−(�m+1

11

(m+1)!∂m+1f∂�m+1 (0, �0) + �1

∂2f∂�∂"

(0, �0))

m�m−11

(1m!

∂mf∂�m

(0, �0)) , (2.9)

�0 = u1, �1 = �1u2, �2 =

⎧⎨⎩ −ΛA′(0)u1 + �2u2, if m = 2

�2u2 + �21u3, if m > 2

(2.10)

for any choice of the mth root of am,1 and where Λ is given in (2.15).

The explicit recursive formulas for computing higher order terms, �k, �k, are given by (2.24)

and (2.25) in Theorem 2. The steps which should be used to determine these higher order

terms are discussed in Remark 1 and an example showing how to calculating �3, �3 using

these steps, when m ≥ 3, is provided.

16

Example

The following example may help to give a better idea of these results. Consider

A (") :=

⎡⎢⎢⎢⎢⎣−1

21 1

2

12

0 −12

−1 1 1

⎤⎥⎥⎥⎥⎦+ "

⎡⎢⎢⎢⎢⎣2 0 −1

2 0 −1

1 0 0

⎤⎥⎥⎥⎥⎦ . (2.11)

Here �0 = 0 is a non-derogatory eigenvalue of A (0) of algebraic multiplicity m = 2. We put

A (0) into the Jordan normal form

U−1A (0)U =

⎡⎢⎢⎢⎢⎣0 1 0

0 0 0

0 0 1/2

⎤⎥⎥⎥⎥⎦ , U =

⎡⎢⎢⎢⎢⎣1 1 1

0 1 1

1 1 0

⎤⎥⎥⎥⎥⎦ , U−1 =

⎡⎢⎢⎢⎢⎣1 −1 0

−1 1 1

1 0 −1

⎤⎥⎥⎥⎥⎦ ,

so that W0 = 1/2. We next define the vectors u1, u2, as the first two columns of the matrix

U ,

u1 :=

⎡⎢⎢⎢⎢⎣1

0

1

⎤⎥⎥⎥⎥⎦ , u2 :=

⎡⎢⎢⎢⎢⎣1

1

1

⎤⎥⎥⎥⎥⎦ .

Next we partition the matrix U−1A′(0)U conformally to the blocks Jm (�0) and W0 of the

matrix U−1A (0)U as such

U−1A′(0)U =

⎡⎢⎢⎢⎢⎣1 −1 0

−1 1 1

1 0 −1

⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣

2 0 −1

2 0 −1

1 0 0

⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣

1 1 1

0 1 1

1 1 0

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣0 ∗ ∗

1 1 ∗

∗ ∗ ∗

⎤⎥⎥⎥⎥⎦ .

17

Here a2,1 = 1, a1,1 = 0, and a2,2 = 1. Then

1 = a2,1 = −∂∂"

det (�I − A (")) ∣(",�)=(0,�0)(∂2

∂�2det(�I−A("))∣(",�)=(0,�0)

2!

) ,

implying that the generic condition (2.1) is true. Define f (", �) := det (�I − A (")) =

�3−2�2"− 12�2 +�"2− 1

2�"+"2 + 1

2". Then there is exactly one convergent Puiseux series for

the perturbed eigenvalues near �0 = 0 and one for their corresponding eigenvectors whose

branches are given by

�ℎ (") = �0 + �1

((−1)ℎ "

12

)+ �2

((−1)ℎ "

12

)2

+∞∑k=3

�k

((−1)ℎ "

12

)kxℎ (") = �0 + �1

((−1)ℎ "

12

)+ �2

((−1)ℎ "

12

)2

+∞∑k=3

�k

((−1)ℎ "

12

)k

for ℎ = 0, 1 and any fixed branch of "12 . Furthermore, the series coefficients up to second

order may be given by

�1 = 1 =√

1 =√a2,1 =

√√√⎷(− ∂f∂"

(0, �0)12!∂2f∂�2

(0, �0)

)∕= 0,

�2 =1

2=a1,1 + a2,2

2=−(�3

113!∂3f∂�3

(0, �0) + �1∂2f∂�∂"

(0, �0))

�1

(12!∂2f∂�2

(0, �0)) ,

�0 =

⎡⎢⎢⎢⎢⎣1

0

1

⎤⎥⎥⎥⎥⎦ , �1 =

⎡⎢⎢⎢⎢⎣1

1

1

⎤⎥⎥⎥⎥⎦ , �2 = −ΛA′(0)u1 + �2u2

by choosing the positive square root of a2,1 = 1 and where Λ is given in (2.15). Here

Λ = U

⎡⎢⎣ Jm (0)∗

(W0 − �0In−m)−1

⎤⎥⎦U−1

18

=

⎡⎢⎢⎢⎢⎣1 1 1

0 1 1

1 1 0

⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣

0 0 0

1 0 0

0 0 (1/2)−1

⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣

1 −1 0

−1 1 1

1 0 −1

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣3 −1 −2

3 −1 −2

1 −1 0

⎤⎥⎥⎥⎥⎦�2 = −ΛA′(0)u1 + �2u2

= −

⎡⎢⎢⎢⎢⎣3 −1 −2

3 −1 −2

1 −1 0

⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣

2 0 −1

2 0 −1

1 0 0

⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣

1

0

1

⎤⎥⎥⎥⎥⎦+1

2

⎡⎢⎢⎢⎢⎣1

1

1

⎤⎥⎥⎥⎥⎦ =1

2

⎡⎢⎢⎢⎢⎣1

1

1

⎤⎥⎥⎥⎥⎦ .

Now compare this to the actual perturbed eigenvalues of our example (2.11) near �0 = 0

and their corresponding eigenvectors

�ℎ (") =1

2"+ (−1)ℎ

1

2"

12 ("+ 4)

12

=(

(−1)ℎ "12

)+

1

2

((−1)ℎ "

12

)2

+∞∑k=3

�k

((−1)ℎ "

12

)k

xℎ (") =

⎡⎢⎢⎢⎢⎣1

0

1

⎤⎥⎥⎥⎥⎦+

⎡⎢⎢⎢⎢⎣1

1

1

⎤⎥⎥⎥⎥⎦�ℎ (")

=

⎡⎢⎢⎢⎢⎣1

0

1

⎤⎥⎥⎥⎥⎦+

⎡⎢⎢⎢⎢⎣1

1

1

⎤⎥⎥⎥⎥⎦(

(−1)ℎ "12

)+

1

2

⎡⎢⎢⎢⎢⎣1

1

1

⎤⎥⎥⎥⎥⎦(

(−1)ℎ "12

)2

+∞∑k=3

�k

((−1)ℎ "

12

)k

for ℎ = 0, 1 and any fixed branch of "12 . We see that indeed our formulas for the Puiseux

series coefficients are correct up to the second order.

19

Comparison to Known Results

There is a fairly large amount of literature on eigenpair perturbation expansions for an-

alytic perturbations of non-selfadjoint matrices with degenerate eigenvalues (e.g. [1, 2, 4,

9, 23, 26, 28, 33, 34, 37, 41, 43, 46–48, 53, 55–58]). However, most of the literature (e.g.

[1, 2, 23, 33, 34, 37, 41, 43, 46–48, 55, 56]) contains results only on the first order expansions

of the Puiseux series or considers higher order terms only in the case of simple or semisimple

eigenvalues. For those works that do address higher order terms for defective eigenvalues

(e.g. [4, 9, 26, 28, 53, 57, 58]), it was found that there did not exist explicit recursive formulas

for all the Puiseux coefficients when the matrix perturbations were non-linear. One of the

purposes and achievements of this paper are the explicit recursive formulas (2.23)–(2.25)

in Theorem 2 which give all the higher order terms in the important case of degenerate

eigenvalues which are non-derogatory, that is, the case in which a degenerate eigenvalue of

the unperturbed matrix has a single Jordan block for its corresponding Jordan structure.

Our theorem generalizes and extends the results of [4, pp. 315–317, (4.96) & (4.97)], [57, pp.

415–418], and [58, pp. 17–20] to non-linear analytic matrix perturbations and makes explicit

the recursive formulas for calculating the perturbed eigenpair Puiseux expansions. Further-

more, in Proposition 8 we give an explicit recursive formula for calculating the polynomials

{rl}l∈ℕ. These polynomials must be calculated in order to determine the higher order terms

in the eigenpair Puiseux series expansions (see (2.25) in Theorem 2 and Remark 1). These

polynomials appear in [4, p. 315, (4.95)], [57, p. 414, (32.24)], and [58, p. 19, (34)] under dif-

ferent notation (compare with Proposition 2.41.ii) but no method is given to calculate them.

As such, Proposition 8 is an important contribution in the explicit recursive calculation of

the higher order terms in the eigenpair Puiseux series expansions.

Another purpose of this paper is to give, in the case of degenerate non-derogatory eigenval-

ues, an easily accessible and quickly referenced list of first and second order terms for the

Puiseux series expansions of the perturbed eigenpairs. When the generic condition (2.1) is

20

satisfied, Corollary 4 gives this list. Now for first order terms there are quite a few papers on

formulas for determining them, see for example [48] which gives a good survey of first order

perturbation theory. But for second order terms, it was difficult to find any results in the

literature similar to and as explicit as Corollary 4 for the case of degenerate non-derogatory

eigenvalues with arbitrary algebraic multiplicity and non-linear analytic perturbations. Re-

sults comparable to ours can be found in [4, p. 316], [57, pp. 415–418], [58, pp. 17-20], and

[53, pp. 37–38, 50–54, 125–128], although it should be noted that in [57, p. 417] the formula

for the second order term of the perturbed eigenvalues contains a misprint.

Overview

Section 2 deals with the generic condition (2.1). We give conditions that are equivalent to the

generic condition in Theorem 1. In §3 we give the main results of this paper in Theorem 2,

on the determination of the Puiseux series with the explicit recursive formulas for calculating

the series coefficients. As a corollary we give the exact leading order terms, up to the second

order, for the Puiseux series coefficients. Section 4 contains the proofs of the results in §2

and §3.

Notation

Let Mn(ℂ) be the set of all n× n matrices with complex entries and ℂn the set of all n× 1

column vectors with complex entries. For a ∈ ℂ, A ∈ Mn(ℂ), and x = [ai,1]ni=1 ∈ ℂn we

denote by a∗, A∗, and x∗, the complex conjugate of a, the conjugate transpose of A, and the

1× n row vector x∗ := [ a∗1,1 ⋅ ⋅ ⋅ a∗n,1 ]. For x, y ∈ ℂn we let ⟨x, y⟩ := x∗y be the standard

inner product. The matrix I ∈Mn(ℂ) is the identity matrix and its jth column is ej ∈ ℂn.

The matrix In−m is the (n−m)× (n−m) identity matrix. Define an m×m Jordan block

21

with eigenvalue � to be

Jm (�) :=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

� 1

. .

. .

. 1

�

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

When the matrix A (") ∈ Mn(ℂ) is analytic at " = 0 we define A′(0) := dAd"

(0) and Ak :=

1k!dkAd"k

(0). Let � := ei2�m .

2.2 The Generic Condition and its Consequences

The following theorem, which is proved in §4, gives conditions which are equivalent to the

generic one (2.1).

Theorem 1 Let A (") be a matrix-valued function having a range in Mn(ℂ) such that its

matrix elements are analytic functions of " in a neighborhood of the origin. Let �0 be an

eigenvalue of the unperturbed matrix A (0) and denote by m its algebraic multiplicity. Then

the following statements are equivalent:

( i) The characteristic polynomial det (�I − A (")) has a simple zero with respect to " at

� = �0 and " = 0, i.e.,

∂

∂"det (�I − A ("))

∣∣(",�)=(0,�0)

∕= 0.

( ii) The characteristic equation, det(�I − A (")) = 0, has a unique solution, " (�), in a

neighborhood of � = �0 with " (�0) = 0. This solution is an analytic function with a

22

zero of order m at � = �0, i.e.,

d0" (�)

d�0

∣∣∣�=�0

= ⋅ ⋅ ⋅ = dm−1" (�)

d�m−1

∣∣∣�=�0

= 0,dm" (�)

d�m

∣∣∣�=�0

∕= 0.

( iii) There exists a convergent Puiseux series whose branches are given by

�ℎ (") = �0 + �1�ℎ"

1m +

∞∑k=2

�k

(�ℎ"

1m

)k,

for ℎ = 0, . . . ,m−1 and any fixed branch of "1m , where � = e

2�mi, such that the values of

the branches give all the solutions of the characteristic equation, for sufficiently small

" and � sufficiently near �0. Furthermore, the first order term is nonzero, i.e.,

�1 ∕= 0.

( iv) The Jordan normal form of A (0) corresponding to the eigenvalue �0 consists of a single

m ×m Jordan block and there exists an eigenvector u0 of A (0) corresponding to the

eigenvalue �0 and an eigenvector v0 of A (0)∗ corresponding to the eigenvalue �∗0 such

that

⟨v0, A′(0)u0⟩ ∕= 0.

2.3 Explicit Recursive Formulas for Calculating the

Perturbed Spectrum

This section contains the main results of this paper presented below in Theorem 2. To begin

we give some preliminaries that are needed to set up the theorem. Suppose that A (") is

a matrix-valued function having a range in Mn(ℂ) with matrix elements that are analytic

23

functions of " in a neighborhood of the origin and �0 is an eigenvalue of the unperturbed

matrix A (0) with algebraic multiplicity m. Assume that the generic condition

∂

∂"det (�I − A ("))

∣∣(",�)=(0,�0)

∕= 0,

is true.

Now, by these assumptions, we may appeal to Theorem 1.(iv) and conclude that the Jordan

canonical form of A(0) has only one m ×m Jordan block associated with �0. Hence there

exists a invertible matrix U ∈ ℂn×n such that

U−1A (0)U =

⎡⎢⎣ Jm (�0)

W0

⎤⎥⎦ , (2.12)

where W0 is a (n−m)× (n−m) matrix such that �0 is not one of its eigenvalues [35, §6.5:

The Jordan Theorem].

We define the vectors u1, . . . , um, v1, . . . , vm ∈ ℂn as the first m columns of the matrix U

and (U−1)∗, respectively, i.e.,

ui := Uei, 1 ≤ i ≤ m, (2.13)

vi :=(U−1

)∗ei, 1 ≤ i ≤ m. (2.14)

And define the matrix Λ ∈Mn(ℂ) by

Λ := U

⎡⎢⎣ Jm (0)∗

(W0 − �0In−m)−1

⎤⎥⎦U−1, (2.15)

where (W0 − �0In−m)−1 exists since �0 is not an eigenvalue ofW0(for the important properties

of the matrix Λ see Section 2.5).

24

Next, we introduce the polynomials pj,i = pj,i (�1, . . . , �j−i+1) in �1,. . . , �j−i+1, for j ≥ i ≥

0, as the expressions

p0,0 := 1, pj,0 := 0, for j > 0,

pj,i (�1, . . . , �j−i+1) :=∑

s1+⋅⋅⋅+si=j1≤s%≤j−i+1

i∏%=1

�s% , for j ≥ i > 0

⎫⎬⎭ (2.16)

and the polynomials rl = rl(�1, . . . , �l) in �1,. . . , �l, for l ≥ 1, as the expressions

r1 := 0, rl(�1, . . . , �l) :=∑

s1+⋅⋅⋅+sm=m+l1≤s%≤l

m∏%=1

�s% , for l > 1 (2.17)

(see Section 2.5 for more details on these polynomials including recursive formulas for their

calculation).

With these preliminaries we can now state the main results of this paper. Proofs of these

results are contained in the next section.

Theorem 2 Let A (") be a matrix-valued function having a range in Mn(ℂ) such that its

matrix elements are analytic functions of " in a neighborhood of the origin. Let �0 be an

eigenvalue of the unperturbed matrix A (0) and denote by m its algebraic multiplicity. Suppose

that the generic condition

∂

∂"det (�I − A ("))

∣∣(",�)=(0,�0)

∕= 0, (2.18)

is true. Then there is exactly one convergent Puiseux series for the �0-group and one for

their corresponding eigenvectors whose branches are given by

�ℎ (") = �0 +∞∑k=1

�k

(�ℎ"

1m

)k(2.19)

25

xℎ (") = �0 +∞∑k=1

�k

(�ℎ"

1m

)k(2.20)

for ℎ = 0, . . . ,m− 1 and any fixed branch of "1m , where � = e

2�mi with

�m1 = ⟨vm, A1u1⟩ = −∂∂"

det (�I − A (")) ∣(",�)=(0,�0)( ∂m

∂�mdet(�I−A("))∣(",�)=(0,�0)

m!

) ∕= 0

(Here A1 denotes dAd"

(0) and the vectors u1 and vm are defined in (2.13) and (2.14)). Fur-

thermore, we can choose

�1 = ⟨vm, A1u1⟩1/m, (2.21)

for any fixed mth root of ⟨vm, A1u1⟩ and the eigenvectors to satisfy the normalization condi-

tions

⟨v1, xℎ (")⟩ = 1, ℎ = 0, ...,m− 1. (2.22)

Consequently, under these conditions �1, �2,. . . and �0, �1, . . . are uniquely determined

and are given by the recursive formulas

�1 = ⟨vm, A1u1⟩1/m =

⎛⎜⎝− ∂∂"

det (�I − A (")) ∣(",�)=(0,�0)( ∂m

∂�mdet(�I−A("))∣(",�)=(0,�0)

m!

)⎞⎟⎠

1/m

(2.23)

�s =

−rs−1 +min{s,m}−1∑

i=0

s−1∑j=i

pj,i

⟨vm−i,

⌊m+s−1−jm ⌋∑k=1

Ak�m+s−1−j−km

⟩m�m−1

1

(2.24)

�s =

⎧⎨⎩s∑i=0

ps,iui+1, if 0 ≤ s ≤ m− 1

m−1∑i=0

ps,iui+1 −s−m∑j=0

j∑k=0

⌊ s−jm ⌋∑l=1

pj,kΛk+1Al�s−j−lm, if s ≥ m

(2.25)

where ui and vi are the vectors defined in (2.13) and (2.14), pj,i and rl are the polynomials

defined in (2.16) and (2.17), ⌊⌋ denotes the floor function, Ak denotes the matrix 1k!dkAd"k

(0),

and Λ is the matrix defined in (2.15).

26

Corollary 3 The calculation of the kth order terms, �k and �k, requires only the matrices

A0, . . . , A⌊m+k−1m ⌋.

Corollary 4 The coefficients of those Puiseux series up to second order are given by

�1 =

(−

∂f∂"

(0, �0)1m!

∂mf∂�m

(0, �0)

)1/m

= ⟨vm, A1u1⟩1/m,

�2 =

⎧⎨⎩−(�m+11

1(m+1)!

∂m+1f

∂�m+1 (0,�0)+�1∂2f∂�∂"

(0,�0)+ 12∂2f

∂"2(0,�0)

)m�m−1

1 ( 1m!

∂mf∂�m

(0,�0)), if m = 1

−(�m+11

1(m+1)!

∂m+1f

∂�m+1 (0,�0)+�1∂2f∂�∂"

(0,�0)

)m�m−1

1 ( 1m!

∂mf∂�m

(0,�0)), if m > 1

=

⎧⎨⎩ ⟨v1, (A2 − A1ΛA1)u1⟩, if m = 1

(vm−1,A1u1)+(vm,A1u2)

m�m−21

, if m > 1,

�0 = u1,

�1 =

⎧⎨⎩ −ΛA1u1, if m = 1

�1u2, if m > 1,

�2 =

⎧⎨⎩(−ΛA2 + (ΛA1)2 − �1Λ2A1

)u1, if m = 1

−ΛA1u1 + �2u2, if m = 2

�2u2 + �21u3, if m > 2

,

where f (", �) := det (�I − A (")).

Remark 1 Suppose we want to calculate the terms �k+1, �k+1, where k ≥ 2, using the

explicit recursive formulas given in the theorem. We may assume we already known or have

calculated

A0, . . . , A⌊m+km ⌋, {rj}

k−1j=1 , {�j}kj=1, {�j}kj=0, {{pj,i}kj=i}ki=0. (2.26)

We need these to calculate �k+1, �k+1 and the steps to do this are indicated by the following

27

arrow diagram:

(2.26)(2.45)→ rk

(2.24)→ �k+1(2.44)→ {pk+1,i}k+1

i=0

(2.25)→ �k+1. (2.27)

After we have followed these steps we not only will have calculated �k+1, �k+1 but we will

now know

A0, . . . , A⌊m+k+1m ⌋, {rj}

kj=1, {�j}k+1

j=1 , {�j}k+1j=0 , {{pj,i}k+1

j=i }k+1i=0 (2.28)

as well. But these are the terms in (2.26) for k+ 1 and so we may repeat the steps indicated

above to calculate �k+2, �k+2.

It is in this way we see how all the higher order terms can be calculated using the results of

this paper.

Example

In order to illustrate these steps we give the following example which recursively calculates

the third order terms for m ≥ 3.

The goal is to determine �3, �3. To do this we follow the steps indicated in the above remark

with k = 2. The first step is to collect the terms in (2.26). Assuming A0, A1 are known then

by (2.16), (2.17), Corollary 4, and Proposition 7 we have

A0, A1, r1 = 0, �1 = ⟨vm, A1u1⟩1/m, �2 = ⟨vm−1,A1u1⟩+⟨vm,A1u2⟩m�m−2

1

,

�0 = u1, �1 = �1u2, �2 = �2u2 + �21u3,

p0,0 = 1, p1,0 = 0, p1,1 = �1, p2,0 = 0, p2,1 = �2, p2,2 = �21.

The next step is to determine r2 using the recursive formula for the rl’s given in (2.45). We

28

find that

r2 =1

2�1

1∑j=1

[(3− j)m− (m+ j)]�3−jrj +m

2�m−2

1

1∑j=1

[(3− j)m− (m+ j)]�3−j�j+1

=m(m− 1)

2�m−2

1 �22.

Now, since r2 is determined, we can use the recursive formula in (2.24) for the �s’s to

calculate �3. In doing so we find that

�3 =

−r2 +min{3,m}−1∑

i=0

2∑j=i

pj,i

⟨vm−i,

⌊m+2−jm ⌋∑k=1

Ak�m+2−j−km

⟩m�m−1

1

=−r2 + p2,1⟨vm−1, A1�0⟩+ p0,0⟨vm, A1�2⟩

m�m−11

+

p2,2⟨vm−2, A1�0⟩+ p1,1⟨vm−1, A1�1⟩m�m−1

1

=−m(m−1)

2�m−2

1 �22 + �2⟨vm−1, A1u1⟩+ ⟨vm, A1(�2u2 + �2

1u3)⟩m�m−1

1

+

�21⟨vm−2, A1u1⟩+ �1⟨vm−1, A1�1u2⟩

m�m−11

=

(3−m

2

)�−1

1 �22 +⟨vm−2, A1u1⟩+ ⟨vm−1, A1u2⟩+ ⟨vm, A1u3⟩

m�m−31

.

Next, since �3 is determined, we can use (2.44) to calculate {p3,i}3i=0. In this case though it

suffices to use Proposition 7 and in doing so we find that

p3,0 = 0, p3,1 = �3, p3,2 = 2�1�2, p3,3 = �31.

Finally, we can compute �3 using the recursive formula in (2.25) for the �s’s. In doing so we

29

find that

�3 =

⎧⎨⎩3∑i=0

p3,iui+1, if m > 3

m−1∑i=0

p3,iui+1 −3−m∑j=0

j∑k=0

⌊ 3−jm ⌋∑l=1

pj,kΛk+1Al�3−j−lm, if m = 3

=

⎧⎨⎩p3,1u2 + p3,2u3 + p3,3u4, if m > 3

2∑i=0

p3,iui+1 − ΛA1�0, if m = 3

=

⎧⎨⎩ �3u2 + 2�1�2u3 + �31u4, if m > 3

�3u2 + 2�1�2u3 − ΛA1u1, if m = 3.

This completes the calculation of the third order terms, �3, �3, when m ≥ 3.

2.4 Proofs

This section contains the proofs of the results of this paper. We begin by proving Theorem

1 of §2 on conditions equivalent to the generic condition. We next follow this up with the

proof of the main result of this paper Theorem 2. We finish by proving the Corollaries 3 and

4.

Proof of Theorem 1

To prove this theorem we will prove the chain of statements (i)⇒(ii)⇒(iii)⇒(iv)⇒(i).

We begin by proving (i)⇒(ii). Define f (", �) := det (�I − A (")) and suppose (i) is true.

Then f is an analytic function of (", �) near (0, �0) since the matrix elements of A (") are

analytic functions of " in a neighborhood of the origin and the determinant of a matrix

is a polynomial in its matrix elements. Also we have f (0, �0) = 0 and ∂f∂"

(0, �0) ∕= 0.

30

Hence by the holomorphic implicit function theorem [30, §1.4 Theorem 1.4.11] there exists

a unique solution, " (�), in a neighborhood of � = �0 with " (�0) = 0 to the equation

f (", �) = 0, which is analytic at � = �0. We now show that " (�) has a zero there of order

m at � = �0. First, the properties of "(�) imply there exists "q ∕= 0 and q ∈ ℕ such that

" (�) = "q (�− �0)q + O((�− �0)q+1), for ∣� − �0∣ << 1. Next, by hypothesis �0 is an

eigenvalue of A (0) of algebraic multiplicity m hence ∂if∖∂�i (0, �0) = 0 for 0 ≤ i ≤ m − 1

but ∂mf∖∂�m (0, �0) ∕= 0. Combining this with the fact that f (0, �0) = 0 and ∂f∂"

(0, �0) ∕= 0

we have

f (", �) = a10"+ a0m (�− �0)m +∑

i+j≥2, i,j∈ℕ(i,j)∕∈{(0,j):j≤m}

aij"i (�− �0)j (2.29)

for ∣"∣ + ∣� − �0∣ << 1, where a10 = ∂f∂"

(0, �0) ∕= 0 and a0m = 1m!

∂mf∂�m

(0, �0) ∕= 0. Then

using the expansions of f (", �) and " (�) together with the identity f (" (�) , �) = 0 for

∣�− �0∣ << 1, we find that q = m and

"m = −a0m

a10

= −1m!

∂m det(�I−A("))∂�m

∣∣∣(�,")=(�0,0)

∂∂"

det (�I − A ("))∣∣(�,")=(�0,0)

. (2.30)

Therefore we conclude that " (�) has a zero of order m at � = �0, which proves (ii).

Next, we prove (ii)⇒(iii). Suppose (ii) is true. The first part of proving (iii) involves inverting

" (�) near " = 0 and � = �0. To do this we expand " (�) in a power series about � = �0 and

find that " (�) = g(�)m where

g(�) = (�− �0)

("m +

∞∑k=m+1

"k (�− �0)k−m)1/m

and we are taking any fixed branch of the mth root that is analytic at "m. Notice that,

for � in a small enough neighborhood of �0, g is an analytic function, g (�0) = 0, and

dgd�

(�0) = "1/mm ∕= 0. This implies, by the inverse function theorem for analytic functions,

that for � in a small enough neighborhood of �0 the analytic function g (�) has an analytic

31

inverse g−1 (") in a neighborhood of " = 0 with g−1 (0) = �0. Define a multivalued function

� ("), for sufficiently small ", by � (") := g−1("

1m

)where by "

1m we mean all branches of

the mth root of ". We know that g−1 is analytic at " = 0 so that for sufficiently small " the

multivalued function � (") is a Puiseux series. And since dg−1

d"(0) =

[dgd�

(�0)]−1 ∕= 0 we have

an expansion

� (") = g−1("

1m

)= �0 + �1"

1m +

∞∑k=2

�k

("

1m

)k.

Now suppose for fixed � sufficiently near �0 and for sufficiently small " that det (�I − A (")) =

0. We want to show this implies � = � (") for one of the branches of the mth root. We know

by hypothesis we must have " = " (�). But as we know this implies that " = " (�) = g(�)m

hence for some branch of the mth root, bm(⋅ ), we have bm(") = bm(g(�)m) = g(�). But

� is near enough to �0 and " is sufficiently small that we may apply the g−1 to both sides

yielding � = g−1 (g(�)) = g−1 (bm(")) = � ("), as desired. Furthermore, all the m branches

�ℎ ("), ℎ = 0, . . . ,m− 1 of � (") are given by taking all branches of the mth root of " so that

�ℎ (") = �0 + �1�ℎ"

1m +

∞∑k=2

�k

(�ℎ"

1m

)k

for any fixed branch of "1m , where � = e

2�mi and

�1 =dg−1

d"(0) =

[dg

d�(�0)

]−1

= "−1/mm ∕= 0, (2.31)

which proves (iii).

Next, we prove (iii)⇒(iv). Suppose (iii) is true. Define the function y (") := �0 ("m).

Then y is analytic at " = 0 and dyd"

(0) = �1 ∕= 0. Also we have for " sufficiently small

det (y (") I − A ("m)) = 0. Consider the inverse of y ("), y−1 (�). It has the property

that 0 = det(y (y−1 (�)) I − A

([y−1 (�)]

m))= det

(�I − A

([y−1 (�)]

m))with y−1 (�0) = 0,

dy−1

d�(�0) = �−1

1 . Define g (�) := [y−1 (�)]m

. Then g has a zero of order m at �0 and

det (�I − A (g (�))) = 0 for � in a neighborhood of �0.

32

Now we consider the analytic matrix A (g (�)) − �I in a neighborhood of � = �0 with the

constant eigenvalue 0. Because 0 is an analytic eigenvalue of it then there exists an analytic

eigenvector, x (�), of A (g (�))− �I corresponding to the eigenvalue 0 in a neighborhood of

�0 such that x (�0) ∕= 0. Hence for � near �0 we have

0 = (A (g (�))− �I)x (�)

=(A(�−m1 (�− �0)m +O

((�− �0)m+1))− (�− �0) I − �0I

)x (�)

= (A (0)− �0I)x (�0) +

((A (0)− �0I)

dx

d�(�0)− x (�0)

)(�− �0) + ⋅ ⋅ ⋅

+

((A (0)− �0I)

dm−1x

d�m−1 (�0)− dm−2x

d�m−2 (�0)

)(�− �0)m−1

+

((A (0)− �0I)

dmx

d�m(�0)− dm−1x

d�m−1 (�0) + �−m1 A′(0)x (�0)

)(�− �0)m

+O((�− �0)m+1) .

This implies that

(A (0)− �0I)x (�0) = 0, (A (0)− �0I)djx

d�j(�0) =

dj−1x

d�j−1 (�0) , for j = 1, . . . ,m− 1,

(A (0)− �0I)dmx

d�m(�0) =

dm−1x

d�m−1 (�0)− �−m1 A′(0)x (�0) . (2.32)

The first m equations imply that x (�0), dxd�

(�0), . . ., dm−1xd�m−1 (�0) is a Jordan chain of length

m generated by dm−1xd�m−1 (�0). Since the algebraic multiplicity of �0 for A (0) is m this implies

that the there is a single m×m Jordan block corresponding to the eigenvalue �0 where we

can take x (�0) , dxd�

(�0) , ..., dm−1xd�m−1 (�0) as a Jordan basis. It follows from basic properties of

Jordan chains that there exists an eigenvector v of A(0)∗ corresponding to the eigenvalue �∗0

such that ⟨v, dm−1xd�m−1 (�0)⟩ = 1. Hence

0 =

⟨(A(0)− �0I)∗v,

dmx

d�m(�0)

⟩(2.32)= 1− �−m1 ⟨v, A′(0)x (�0)⟩

33

implying that ⟨v, dAd"

(0)x (�0)⟩ = �m1 ∕= 0. Therefore we have shown that the Jordan normal

form of A (0) corresponding to the eigenvalue �0 consists of a single m×m Jordan block and

there exists an eigenvector u of A (0) corresponding to the eigenvalue �0 and an eigenvector

v of A (0)∗ corresponding to the eigenvalue �∗0 such that ⟨v,A′(0)u⟩ ∕= 0. This proves (iv).

Finally, we show (iv)⇒(i). Suppose (iv) is true. We begin by noting that since

det (�0I − A (")) = (−1)n det ((A (0)− �0I) + A′(0)") + o (")

it suffices to show that

Sn−1 :=d

d"det ((A (0)− �0I) + A′(0)")

∣∣"=0∕= 0. (2.33)

We will use the result from [24, Theorem 2.16] to prove (2.33). Let A (0)− �0I = Y ΣX∗ be

a singular-value decomposition of the matrix A (0) − �0I where X, Y are unitary matrices

and Σ = diag(�1, . . . , �n−1, �n) with �1 ≥ . . . ≥ �n−1 ≥ �n ≥ 0 (see [35, §5.7, Theorem

2]). Now since the Jordan normal form of A (0) corresponding to the eigenvalue �0 consists

of a single Jordan block this implies that rank of A (0) − �0I is n − 1. This implies that

�1 ≥ . . . ≥ �n−1 > �n = 0, u = Xen is an eigenvalue of A (0) corresponding to the eigenvalue

�0, v = Y en is an eigenvalue of A (0) corresponding to the eigenvalue �∗0, and there exist

nonzero constants c1, c2 such that u = c1u0 and v = c2v0.

Now using the result of [24, Theorem 2.16] for (2.33) we find that

Sn−1 = det (Y X∗)∑

1≤i1<⋅⋅⋅<in−1≤n

�i1 ⋅ ⋅ ⋅ �in−1 det(

(Y ∗A′(0)X)i1...in−1

),

where (Y ∗A′(0)X)i1...in−1is the matrix obtained from Y ∗A′(0)X by removing rows and

34

columns i1 . . . in−1. But since �n = 0 and

(Y ∗A′(0)X)1...(n−1) = e∗nY∗A′(0)Xen = (v, A′(0)u) = c∗2c1 (v0, A

′(0)u0) ∕= 0

then Sn−1 = det (Y X∗)n−1∏j=1

�jc∗2c1 (v0, A

′(0)u0) ∕= 0. This completes the proof.

Proof of Theorem 2

We begin by noting that our hypotheses imply that statements (ii), (iii), and (iv) of Theorem

1 are true. In particular, statement (iii) implies that there is exactly one convergent Puiseux

series for the �0-group whose branches are given by

�ℎ (") = �0 + �1�ℎ"

1m +

∞∑k=2

�k

(�ℎ"

1m

)k,

for ℎ = 0, . . . ,m− 1 and any fixed branch of "1m , where � = e

2�mi and �1 ∕= 0. Then by well

known results [4, §6.1.7, Theorem 2], [28, §II.1.8] there exists a convergent Puiseux series for

the corresponding eigenvectors whose branches are given by

xℎ (") = �0 +∞∑k=1

�k

(�ℎ"

1m

)k,

for ℎ = 0, . . . ,m−1, where �0 is an eigenvector of A0 = A (0) corresponding to the eigenvalue

�0. Now if we examine the proof of (ii)⇒(iii) in Theorem 1 we see by equation (2.31) that

�m1 = "−1m , where "m is given in equation (2.30) in the proof of (i)⇒(iii) for Theorem 1. Thus

we can conclude that

�m1 = −∂∂"

det (�I − A (")) ∣(",�)=(0,�0)( ∂m

∂�mdet(�I−A("))∣(",�)=(0,�0)

m!

) ∕= 0. (2.34)

35

Choose any mth root of ⟨vm, A1u1⟩ and denote it by ⟨vm, A1u1⟩1/m. By (2.34) we can just

reindexing the Puiseux series (2.19) and (2.20) and assume that

�1 =

⎛⎜⎝− ∂∂"

det (�I − A (")) ∣(",�)=(0,�0)( ∂m

∂�mdet(�I−A("))∣(",�)=(0,�0)

m!

)⎞⎟⎠

1/m

.

Next, we wish to prove that we can choose the perturbed eigenvectors (2.20) to satisfy the

normalization conditions (2.22). But this follows by Theorem 1 (iv) and the fact �0 is an

eigenvector of A(0) corresponding to the eigenvalue �0 since then ⟨v1, �0⟩ ∕= 0 and so we may

take xℎ(")(v1,xℎ("))

, for ℎ = 0, . . . ,m − 1, to be the perturbed eigenvectors in (2.20) that satisfy

the normalization conditions (2.22).

Now we are ready to begin showing that {�s}∞s=1, {�s}∞s=0 are given by the recursive formulas

(2.23)-(2.25). The first key step is proving the following:

(A0 − �0I) �s = −s∑

k=1

(A k

m− �kI

)�s−k, for s ≥ 1, (2.35)

�0 = u1, �s = Λ (A0 − �0I) �s, for s ≥ 1, (2.36)

where we define A km

:= 0, if km∕∈ ℕ.

The first equality holds since in a neighborhood of the origin

0 = (A (")− �0 (") I)x0 (") =∞∑s=0

(s∑

k=0

(A k

m− �kI

)�s−k

)"sm .

The second equality will be proven once we show �0 = u1 and �s ∈ S := span{Uei∣2 ≤

i ≤ n}, for s ≥ 1, where U is the matrix from (2.12). This will prove (2.36) because

Λ (A0 − �0I) acts as the identity on S by Proposition 6.i. But these follow from the facts that

36

S = {x ∈ ℂn∣⟨v1, x⟩ = 0} and the normalization conditions (2.22) imply that ⟨v1, �0⟩ = 1

and ⟨v1, �s⟩ = 0, for s ≥ 1.

The next key step in this proof is the following lemma:

Lemma 5 For all s ≥ 0 the following identity holds

(A0 − �0I) �s =

⎧⎨⎩s∑i=0

ps,iui, for 0 ≤ s ≤ m− 1

m∑i=0

ps,iui −s−m∑j=0

j∑k=0

⌊ s−jm ⌋∑l=1

pj,kΛkAl�s−j−lm, for s ≥ m

(2.37)

where we define u0 := 0.

Proof. The proof is by induction on s. The statement is true for s = 0 since p0,0u0 =

0 = (A0 − �0I) �0. Now suppose it was true for all r with 0 ≤ r ≤ s for some nonnegative

integer s. We will show the statement is true for s+ 1 as well.

Suppose s + 1 ≤ m − 1 then (A0 − �0I) �r =∑r

i=0 pr,iui for 0 ≤ r ≤ s and we must show

that (A0 − �0I) �s+1 =∑s+1

i=0 ps+1,iui. Well, for 1 ≤ r ≤ s,

�r(2.36)= Λ (A0 − �0I) �r =

r∑i=0

pr,iΛui(2.40)=

r∑i=1

pr,iui+1. (2.38)

Hence the statement is true if s+ 1 ≤ m− 1 since

(A0 − �0I) �s+1

(2.35)= −

s+1∑k=1

(A k

m− �kI

)�s+1−k

(2.38)=

s+1∑k=1

s+1−k∑i=0

�kps+1−k,iui+1

(2.46)=

s∑i=0

(s+1−i∑k=1

�kps+1−k,i

)ui+1

(2.43)=

s+1∑i=0

ps+1,iui.

Now suppose that s+ 1 ≥ m. The proof is similar to what we just proved. By the induction

37

hypothesis (2.37) is true for 1 ≤ r ≤ s and �r(2.36)= Λ (A0 − �0I) �r thus

�r(2.40)=

⎧⎨⎩r∑i=0

pr,iui+1, for 0 ≤ r ≤ m− 1

m−1∑i=0

pr,iui+1 −r−m∑j=0

j∑k=0

⌊ r−jm ⌋∑l=1

pj,kΛk+1Al�r−j−lm, for r ≥ m.

(2.39)

Hence we have

(A0 − �0I) �s+1

(2.35)= −

s+1∑k=1

(A k

m− �kI

)�s+1−k

(2.39)= −

⌊ s+1m ⌋∑l=1

Al�s+1−lm +s+1−m∑k=1

m−1∑i=0

�kps+1−k,iui+1

−s+1−m∑k=1

s+1−k−m∑j=0

j∑i=0

⌊ s+1−k−jm ⌋∑l=1

�kpj,iΛi+1Al�s+1−k−j−lm

+s+1∑

k>s+1−m

s+1−k∑i=0

�kps+1−k,iui+1

(2.46)= −

⌊ s+1m ⌋∑l=1

Al�s+1−lm +m−1∑i=0

(s+1−i∑k=1

�kps+1−k,i

)ui+1

−s+1−m∑k=1

s+1−k−m∑j=0

j∑i=0

⌊ s+1−k−jm ⌋∑l=1

�kpj,iΛi+1Al�s+1−k−j−lm

(2.43)= −

⌊ s+1m ⌋∑l=1

Al�s+1−lm +m∑i=0

ps+1,iui

−s+1−m∑k=1

s+1−k−m∑j=0

j∑i=0

⌊ s+1−k−jm ⌋∑l=1

�kpj,iΛi+1Al�s+1−k−j−lm.

Now let ak,j,i :=⌊ s+1−k−j

m ⌋∑l=1

�kpj,iΛi+1Al�s+1−k−j−lm. Then using the sum identity

s+1−m∑k=1

s+1−k−m∑j=0

j∑i=0

ak,j,i(2.46)=

s−m∑j=0

s+1−j−m∑k=1

j∑i=0

ak,j,i =s−m∑j=0

j∑i=0

s+1−j−m∑k=1

ak,j,i

(2.48)=

s−m∑i=0

s−m∑j=i

s+1−j−m∑k=1

ak,j,i(2.49)=

s−m∑i=0

s+1−m∑q=i+1

q−i∑k=1

ak,q−k,i

38

we can concluded that

(A0 − �0I) �s+1 = −⌊ s+1m ⌋∑l=1

Al�s+1−lm +m∑i=0

ps+1,iui

−s−m∑i=0

s+1−m∑q=i+1

q−i∑k=1

⌊ s+1−qm ⌋∑l=1

�kpq−k,iΛi+1Al�s+1−q−lm

= −⌊ s+1m ⌋∑l=1

Al�s+1−lm +m∑i=0

ps+1,iui

−s−m∑i=0

s+1−m∑q=i+1

⌊ s+1−qm ⌋∑l=1

(q−i∑k=1

�kpq−k,i

)Λi+1Al�s+1−q−lm

(2.43)= −

⌊ s+1m ⌋∑l=1

Al�s+1−lm +m∑i=0

ps+1,iui

−s−m∑i=0

s+1−m∑q=i+1

⌊ s+1−qm ⌋∑l=1

pq,i+1Λi+1Al�s+1−q−lm

(2.47)= −

⌊ s+1m ⌋∑l=1

Al�s+1−lm +m∑i=0

ps+1,iui

−s+1−m∑q=1

q−1∑i=0

⌊ s+1−qm ⌋∑l=1

pq,i+1Λi+1Al�s+1−q−lm

(2.16)=

m∑i=0

ps+1,iui −s+1−m∑j=0

j∑k=0

⌊ s+1−jm ⌋∑l=1

pj,kΛkAl�s+1−j−lm.

But this is the statement we needed to prove for s + 1 ≥ m. Therefore by induction the

statement (2.37) is true for all s ≥ 0 and the lemma is proved.

The lemma above is the key to prove the recursive formulas for �s and �s as given by

(2.23)-(2.25). First we prove that �s is given by (2.25). For s = 0 we have already shown

�0 = u1 = p0,0u1. So suppose s ≥ 1. Then by (2.36) and (2.37) we find that

�s(2.40)=

⎧⎨⎩s∑i=0

ps,iui+1, if 0 ≤ s ≤ m− 1

m−1∑i=0

ps,iui+1 −s−m∑j=0

j∑k=0

⌊ s−jm ⌋∑l=1

pj,kΛk+1Al�s−j−lm, if s ≥ m.

This proves that �s is given by (2.25).

39

Next we will prove that �s is given by (2.23) and (2.24). We start with s = 1 and prove �1

is given by (2.23). First, (A0 − �0I)∗vm = 0 and ⟨vm, ui⟩ = �m,i hence

0 = ⟨vm, (A0 − �0I)�m⟩(2.37)=

⟨vm,

m∑i=0

pm,iui − A1u1

⟩(2.42)= �m1 − ⟨vm, A1u1⟩

so that �m1 = ⟨vm, A1u1⟩. This and identity (2.34) imply that formula (2.23) is true.

Finally, suppose that s ≥ 2. Then (A0 − �0I)∗vm = 0 and ⟨vm, ui⟩ = �m,i implies

0 = ⟨vm, (A0 − �0I)�m+s−1⟩

(2.37)=

⟨vm,

m∑i=0

pm+s−1,iui −s−1∑j=0

j∑k=0

⌊m+s−1−jm ⌋∑l=1

pj,kΛkAl�m+s−1−j−lm

⟩(2.48)= pm+s−1,m −

s−1∑k=0

s−1∑j=k

pj,k

⟨(Λ∗)k vm,

⌊m+s−1−jm ⌋∑l=1

Al�m+s−1−j−lm

⟩(2.41)= rs−1 +m�m−1

1 �s −s−1∑i=0

s−1∑j=i

pj,i

⟨(Λ∗)i vm,

⌊m+s−1−jm ⌋∑k=1

Ak�m+s−1−j−km

⟩

Therefore with this equality, the fact �1 ∕= 0, and Proposition 6.iii, we can solve for �s and

we will find that it is given by (2.24). This completes the proof.

Proof of Corollaries 3 and 4

Both corollaries follow almost trivially now. To prove Corollary 3, we just examine the recur-

sive formulas (2.23)-(2.25) in Theorem 2 to see that �k, �k requires only A0, . . . , A⌊m+k−1m ⌋.

To prove Corollary 4, we use Proposition 7 to show that

p0,0 = 1, p1,0 = p2,0 = 0, p1,1 = �1, p2,1 = �2, p2,2 = �21

40

and then from this and (2.23)-(2.25) we get the desired result for �1, �2, �0, �1, �2 in terms

of A0, A1, A2. The last part to prove is the formula for �2 in terms of f (", �) and its partial

derivatives. But the formula follows from the series representation of f (", �) in (2.29) and

�0 (") in (2.19) since, for " in a neighborhood of the origin,

0 = f (", �0 ("))

= (a10 + a0mpm.m) "

+

⎧⎨⎩ (a01p2,1 + a02p2,2 + a11p1,1 + a20p00) "2, for m = 1

(a0mpm+1,m + a0m+1pm+1,m+1 + a11p1,1) "m+1m , for m > 1

+O("m+2m

)

which together with Proposition 7 implies the formula for �2.

2.5 Auxiliary Results

Properties of the Matrix Λ

The fundamental properties of the matrix Λ defined in (2.15) which are needed in this paper

are given in the following proposition:

Proposition 6 (i) We have Λ (A0 − �0I)Ue1 = 0, Λ (A0 − �0I)Uei = Uei, for 2 ≤ i ≤

n,

(ii) For 1 ≤ i ≤ m− 1 we have

Λum = 0, Λui = ui+1 (2.40)

(iii) Λ∗v1 = 0, and Λ∗vi = vi−1, for 2 ≤ i ≤ m.

41

Proof. i. Using the fact Jm (0)∗ Jm (0) = diag[0, Im−1], (2.12), and (2.15) we find by block

multiplication that U−1Λ (A0 − �0I)U = diag[0, In−1]. This implies the result.

ii. & iii. The results follow from the definition of ui, vi in (2.13), (2.14) and the fact

(U−1ΛU)∗ = U∗Λ∗(U−1

)∗=

⎡⎢⎣ Jm (0) [(W0 − �0In−m)−1]∗

⎤⎥⎦ .

Properties of the Polynomials in the Recursive Formulas

This appendix contains two propositions. The first proposition gives fundamental identities

that help to characterize the polynomials {pj,i}∞j=i and {rl}l∈ℕ in (2.16) and (2.17). The

second proposition gives explicit recursive formulas to calculate these polynomials.

We may assume∑∞

j=1 �jzj is a convergent Taylor series and �1 ∕= 0.

Proposition 7 The polynomials {pj,i}∞j=i and {rl}l∈ℕ have the following properties:

(i)∞∑j=i

pj,izj =

(∞∑j=1

�jzj

)i

, for j ≥ i ≥ 0.

(ii) For l ≥ 1 we have

rl = pm+l,m −m�m−11 �l+1. (2.41)

(iii) pj,1 = �j, for j ≥ 1.

(iv) For j ≥ 0 we have

pj,j = �j1. (2.42)

42

(v) pj+1,j = j�j−11 �2, for j > 0.

(vi) For j ≥ i > 0 we have

j−i+1∑q=1

�qpj−q,i−1 = pj,i. (2.43)

Proof. i. For i ≥ 0,

(∞∑j=1

�jzj

)i

=∞∑s1=1

⋅ ⋅ ⋅∞∑si=1

(i∏

%=1

�s%

)zs1+⋅⋅⋅+si =

∞∑j=i

pj,izj.

ii. Let l ≥ 1. Then by definition of pm+l,m we have

pm+l,m =∑

s1+⋅⋅⋅+sm=m+l1≤s%≤l+1

∃%∈{1,...,m} such that s%=l+1

m∏%=1

�s% +∑

s1+⋅⋅⋅+sm=m+l1≤s%≤l+1

/∃%∈{1,...,m} such that s%=l+1

m∏%=1

�s%

= m�m−11 �l+1 + rl.

iii. For j ≥ 1 we have pj,1 =∑

s1=j1≤s%≤j

∏1%=1 �s% = �j.

iv. For j ≥ 0, p0,0 = 1 and pj,j =∑

s1+⋅⋅⋅+sj=j1≤s%≤1

∏j%=1 �s% =

∏j%=1 �1 = �j1.

v. For j > 0, pj+1,j =∑

s1+⋅⋅⋅+sj=j+11≤s%≤2

∏j%=1 �s% =

∑j%=1 �

j−11 �2 = j�j−1

1 �2.

vi. It follows by

∞∑j=i

pj,izj (i)

=∞∑

j=i−1

pj,i−1zj

∞∑j=1

pj,1zj =

∞∑j=i

(j−i+1∑q=1

pj−q,i−1pq,1

)zj.

This next proposition gives explicit recursive formulas to calculate the polynomials {pj,i}∞j=i

and {rl}l∈ℕ.

Proposition 8 For each i ≥ 0, the sequence of polynomials, {pj,i}∞j=i, is given by the recur-

43

sive formula

pi,i = �i1, pj,i =1

(j − i)�1

j−1∑k=i

[(j + 1− k)i− k]�j+1−kpk,i, for j > i. (2.44)

Furthermore, the polynomials {rl}l∈ℕ are given by the recursive formula:

r1 = 0, rl =1

l�1

l−1∑j=1

[(l + 1− j)m− (m+ j)]�l+1−jrj+ (2.45)

m

l�m−2

1

l−1∑j=1

[(l + 1− j)m− (m+ j)]�l+1−j�j+1, for l > 1.

Proof. We begin by showing (2.44) is true. For i = 0, (2.44) follows from the definition

of the pj,0. If i > 0 then by (2.42) and [20, (1.1) & (3.2)] it follows that (2.44) is true. Lets

now prove (2.45). From (2.41) and (2.44), we have

rl = pm+l,m −m�m−11 �l+1

=1

l�1

m+l−1∑k=m

[(m+ l + 1− k)m− k]�m+l+1−kpk,m −m�m−11 �l+1

(2.42)=

1

l�1

m+l−1∑k=m+1

[(m+ l + 1− k)m− k]�m+l+1−kpk,m

=1

l�1

l−1∑j=1

[(l + 1− j)m− (m+ j)]�l+1−jpm+j,m

(2.41)=

1

l�1

l−1∑j=1

[(l + 1− j)m− (m+ j)]�l+1−j(rj +m�m−1

1 �j+1

)=

1

l�1

l−1∑j=1

[(l + 1− j)m− (m+ j)]�l+1−jrj+

m

l�m−2

1

l−1∑j=1

[(l + 1− j)m− (m+ j)]�l+1−j�j+1,

for l > 1. This completes the proof.

44

Sum Identities

These double sum identities are used in the proof of Theorem 3.1

d∑x=c

d−x∑y=0

ax,y =d−c∑y=0

d−y∑x=c

ax,y, (2.46)

d−1∑x=0

d∑y=x+1

ax,y =d∑y=1

y−1∑x=0

ax,y, (2.47)

d∑x=0

x∑y=0

ax,y =d∑y=0

d∑x=y

ax,y, (2.48)

d−1∑y=c

d−y∑x=1

ax,y =d∑

q=c+1

q−c∑x=1

ax,q−x. (2.49)

45

Chapter 3

Spectral Perturbation Theory for

Holomorphic Matrix Functions

In this section we consider holomorphic matrix-valued functions of one and two variables.

A holomorphic matrix-valued function of one variable will be referred to as a holomorphic

matrix function. Our goal is to give some basic facts on the spectral theory and spectral

perturbation theory for holomorphic matrix functions that will be needed in this thesis. The

exposition we give here is drawn from many sources, in particular, [44], [29, Appendix A], and

[19] for the spectral theory and [4, Appendix A], [23], and [34] for the spectral perturbation

theory.

3.1 On Holomorphic Matrix-Valued Functions of One

Variable

We introduce the following notation and convention for this section:

46

1. O(�0) :=∪r>0

O(B(�0, r),ℂ) – the set of scalar functions holomorphic at �0 ∈ ℂ.

2. Om×n(�0) :=∪r>0

O(B(�0, r),Mm,n(ℂ)) – the set of m × n matrix-valued functions

holomorphic at �0 ∈ ℂ.

3. As convention we identify O1×1(�0) with O(�0).

4. If f1, f2 ∈ Om×n(�0) then we use the notation f1 = f2 to mean there exists an r > 0

such that f1, f2 ∈ O(B(�0, r),Mm,n(ℂ)) and f1(�) = f2(�) for ∣�− �0∣ < r.

5. As is known, if f ∈ Om×n(�0) then f(�) is analytic at � = �0, i.e., f(�) is infinitely

differentiable at � = �0 and, denoting its jth derivative at � = �0 by f (j)(�0), there

exists an r > 0 such that

f(�) =∞∑j=0

1

j!f (j)(�0)(�− �0)j, ∣�− �0∣ < r,

where the power series on the right converges absolutely in the Mm,n(ℂ) norm to f(�)

for ∣�− �0∣ < r.

We begin with the definition of the spectrum of a holomorphic matrix function and the

important notion of spectral equivalence for holomorphic matrix functions.

Definition 2 If F ∈ O(U,Mn(ℂ)), where U is an open connected set in ℂ, then the spec-

trum of F , denoted by �(F ), is the subset of U such that the matrix F (�) is not invertible,

i.e.,

�(F ) := {� ∈ U ∣ detF (�) = 0}. (3.1)

A point �0 ∈ U is called an eigenvalue of F provided �0 ∈ �(F ).

Definition 3 Two holomorphic matrix functions F,G ∈ On×n(�0) are called equivalent

47

at �0 provided there exists N,M ∈ On×n(�0) and an r > 0 such that N(�0),M(�0) are

invertible matrices and

F (�) = N(�)G(�)M(�), (3.2)

for every � ∈ B(�0, r).

For the local spectral theory of holomorphic matrix functions the follows theorem plays a

central role:

Theorem 9 (Local Smith Form) Let F ∈ On×n(�0) with �0 ∈ �(F ). Then F is equiva-

lent at �0 to a unique diagonal matrix G ∈ On×n(�0) with

G(�) = diag{0, . . . , 0, (�− �0)m�+1 , . . . , (�− �0)mg , 1, . . . , 1}, (3.3)

where 0 ≤ � ≤ g ≤ n and the number of zeros down the diagonal is �, when � = g there are

no (�−�0) terms appearing in the diagonal, when � < g we have m�+1 ≥ . . . ≥ mg ≥ 1 with

{m�+j}gj=1 ⊆ ℕ, and when g = n there are no ones appearing in the diagonal.

Proof. This existence is proven in [29, pp. 414–415, Theorem A.6.1] and uniqueness follows

from [19, p. 106, Theorem 4.3.1].

Definition 4 The matrix G in (3.3) is called the local Smith form of F corresponding to

the eigenvalue �0.

Now its obvious from the theorem that

g = dim(ker(F (�0))), (3.4)

� = dim(ker(F (�))), 0 < ∣�− �0∣ ≪ 1, (3.5)

48

and, since det(F ) ∈ O(�0),

� = 0⇔ det(F ) ∕= 0. (3.6)

But an interpretation of the numbers {m�+j}gj=1 is not as straight forward. Thus we now

give an important example showing how to interpret this theorem in a special case to the

structure of the Jordan normal form of a matrix corresponding to an eigenvalue. Moreover,

the following example will help to motivate the discussion in the next section.

Example 1. Let A ∈Mn(ℂ) and define F (�) := A−�In for � ∈ ℂ. Let �0 be an eigenvalue

of A. Then the hypotheses of Theorem 9 are satisfied and so F is equivalent at �0 to a

diagonal matrix

G(�) = diag{(�− �0)m1 , . . . , (�− �0)mg , 1, . . . , 1}. (3.7)

where m1 ≥ ⋅ ⋅ ⋅ ≥ mg ≥ 1 and {mj}gj=1 ⊆ ℕ. The numbers g, m1, . . . ,mg, and m :=

m1 + ⋅ ⋅ ⋅ + mg are called the geometric multiplicity, partial multiplicities, and algebraic

multiplicity, respectively, of the eigenvalue �0 of the matrix A. It follows from [18, §2.2] and

[18, p. 657, §A.3, Theorem A.3.4] that the Jordan normal form of the matrix A corresponding

to the eigenvalue �0 consists of g Jordan blocks of order m1, . . . ,mg, respectively, with

eigenvalue �0, that is, there exists matrices S ∈ Mn(ℂ) and W0 ∈ Mn−m(ℂ) such that S

is an invertible matrix, �0 is not an eigenvalue of the matrix W0, and S−1AS is the block

diagonal matrix

S−1AS = diag{Jm1(�0), . . . , Jmg(�0),W0}, (3.8)

49

where Jmj(�0) is the mj ×mj Jordan block given by

Jmj(�0) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�0 1

. .

. .

. 1

�0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, for j = 1, . . . , g, (3.9)

where all unmarked elements are zeros.

In the next section we use Theorem 9 and this example to analogously define for an eigen-

value of a holomorphic matrix function, its geometric multiplicity, partial multiplicities, and

algebraic multiplicity. We also introduce the notion of a Jordan chain for a holomorphic

matrix function and again use this example to compare it with the standard definition of a

Jordan chain of a matrix (see [35, §6.3] for the standard definitions in the matrix case).

3.1.1 Local Spectral Theory of Holomorphic Matrix Functions

For holomorphic matrix functions, a local spectral theory can be given based on the local

Smith form. We now give a brief exposition of this theory.

Definition 5 Let F ∈ On×n(�0). Then ker(F (�0)) is called the eigenspace of F at �0.

If ' ∈ ker(F (�0)) and ' ∕= 0 then ' is called an eigenvector of F corresponding to the

eigenvalue �0.

Definition 6 If F ∈ On×n(�0) and �0 is an eigenvalue F then we say �0 is an eigenvalue

of F of finite (infinite) algebraic multiplicity if det(F ) ∕= 0 (det(F ) = 0).

Definition 7 Let F be any holomorphic matrix function such that F ∈ On×n(�0) and �0 an

50

eigenvalue of F of finite algebraic multiplicity. By Theorem 9 and (3.6) there exists unique

numbers g,m1, . . . ,mg ∈ ℕ satisfying m1 ≥ ⋅ ⋅ ⋅ ≥ mg such that the local Smith form of F

corresponding to the eigenvalue �0 is

G(�) = diag{(�− �0)m1 , . . . , (�− �0)mg , 1, . . . , 1}. (3.10)

The positive integers g, m1, . . . ,mg, and m := m1 + ⋅ ⋅ ⋅ + mg are called the geometric

multiplicity, partial multiplicities, and algebraic multiplicity, respectively, of the

eigenvalue �0 of F . We say �0 is a semisimple eigenvalue of F if m1 = ⋅ ⋅ ⋅ = mg = 1.

The following proposition gives a simple but important characterization of the geometric

and algebraic multiplicity.

Proposition 10 Suppose F ∈ On×n(�0) and �0 is an eigenvalue of F of finite algebraic

multiplicity. Let g and m be the geometric multiplicity and algebraic multiplicity of the

eigenvalue �0 of F . Then g is the dimension of ker(F (�0)) and m is the order of the zero of

the function det(F (�)) at � = �0.

Proof. Let m1 ≥ ⋅ ⋅ ⋅ ≥ mg be the partial multiplicities of the eigenvalue �0 of F so

that m = m1 + ⋅ ⋅ ⋅ + mg. By Theorem 9 we know that F is equivalent at �0 to its local

Smith form. Thus there exists N,M ∈ On×n(�0) and an r > 0 such that N(�0),M(�0) are

invertible matrices and

F (�) = N(�) diag{(�− �0)m1 , . . . , (�− �0)mg , 1, . . . , 1}M(�),

for every � ∈ B(�0, r). It follows from this representation that the first g columns of the

matrix M(�0)−1 are a basis for ker(F (�0)) so that g = dim(ker(F (�0))). Moreover, since

51

det(N), det(M) ∈ O(�0) with det(N(�0)), det(M(�0)) ∕= 0, we have

det(F (�)) = det(N(�) diag{(�− �0)m1 , . . . , (�− �0)mg , 1, . . . , 1}M(�))

= (�− �0)m det(N(�) det(M(�))

= (�− �0)m det(N(�0)) det(M(�0)) +O((�− �0)m+1), as �→ �0.

Hence m is the order of the zero of the function det(F (�)) at � = �0. This completes the

proof.

Definition 8 Let F be any holomorphic matrix function such that F ∈ On×n(�0) and �0 an

eigenvalue of F . A sequence {'j}l−1j=0 ⊆ ℂn, where l ∈ ℕ, is called a Jordan chain of F of

length l corresponding to the eigenvalue �0 (associated with the eigenvector '0) if

'0 ∕= 0 and

j∑ℎ=0

1

ℎ!F (ℎ)(�0)'j−ℎ = 0, j = 0, . . . , l − 1. (3.11)

The maximal length of a Jordan chain of F corresponding to the eigenvalue �0 and associated

with '0 is called the multiplicity of the eigenvector '0 and is denoted by m('0), i.e.,

m('0) := sup{l ∈ ℕ : {'j}l−1j=0 satisfies (3.11)}. (3.12)

Example 2. Let A ∈ Mn(ℂ) and define F (�) := A − �In for � ∈ ℂ. Then by Definition

5, �0 is an eigenvalue of A if and only if �0 is an eigenvalue of F . We will now show that

a sequence of vectors {'j}l−1j=0 ⊆ ℂn is a Jordan chain of A of length l corresponding to the

eigenvalue �0 (see [35, p. 230, §6.3] for definition) if and only if {'j}l−1j=0 is a Jordan chain of

F of length l corresponding to the eigenvalue �0.

52

Suppose �0 is an eigenvalue of A and {'j}l−1j=0 is a Jordan chain of A of length l corresponding

to the eigenvalue �0, i.e.,

'0 ∕= 0, (A− �0In)'j = 'j−1, j = 0, . . . , l − 1

where '−1 := 0. Then '0 ∕= 0 and

j∑ℎ=0

1

ℎ!F (ℎ)(�0)'j−ℎ = F (0)(�0)'j + F (1)(�0)'j−1 = (A− �0In)'j − 'j−1 = 0

for j = 0, . . . , l− 1. This implies by Definition 8 that the sequence {'j}l−1j=0 is a Jordan chain

of F of length l corresponding to the eigenvalue �0.

Conversely, if {'j}l−1j=0 is a Jordan chain of F of length l corresponding to the eigenvalue �0

then by Definition 8 this sequence satisfies

'0 ∕= 0 and 0 =

j∑ℎ=0

1

ℎ!F (ℎ)(�0)'j−ℎ = (A− �0In)'j − 'j−1, j = 0, . . . , l − 1,

where '−1 := 0. This implies {'j}l−1j=0 ⊆ ℂn is a Jordan chain of A of length l corresponding

to the eigenvalue �0.

Definition 9 Let F ∈ On×n(�0). A vector-valued function ' ∈ On×1(�0) is called a gen-

erating function of order l for F at the eigenvalue �0, where l ∈ ℕ, if '(�0) ∕= 0 and the

function F' ∈ On×1(�0) has a zero of order ≥ l at �0, i.e.,

'(�0) ∕= 0 and F (�)'(�) = O((�− �0)l), as �→ �0 (3.13)

Proposition 11 Let F ∈ On×n(�0). Then ' ∈ On×1(�0) is a generating function of order l

53

for F at the eigenvalue �0 if and only if �0 is an eigenvalue F and the sequence

'j :=1

j!'(j)(�0), j = 0, . . . , l − 1 (3.14)

is a Jordan chain of F of length l corresponding to the eigenvalue �0.

Proof. This statement and its proof can be found in [44, p. 57, §II.11, Lemma 11.3]. The

result follows from the fact if F ∈ On×n(�0) and ' ∈ On×1(�0) then F' ∈ On×1(�0) and

hence

F (�)'(�) =∞∑j=0

1

j!F (j)(�0)(�− �0)j

∞∑j=0

1

j!'(j)(�0)(�− �0)j

=∞∑j=0

j∑ℎ=0

1

ℎ!F (ℎ)(�0)

1

(j − ℎ)!'(j−ℎ)(�0)

=l−1∑j=0

j∑ℎ=0

1

ℎ!F (ℎ)(�0)

1

(j − ℎ)!'(j−ℎ)(�0) +O((�− �0)l), as �→ �0.

Definition 10 Let F ∈ On×n(�0) such that �0 an eigenvalue of F of finite algebraic mul-

tiplicity. We say {'i,j}mj−1i=0 , j = 1, . . . , g is a canonical system of Jordan chains of F

corresponding to the eigenvalue �0 if the following three conditions are satisfied:

1. The numbers g, m1, . . . ,mg, and m := m1 + ⋅ ⋅ ⋅ + mg are the geometric multiplicity,

partial multiplicities, and algebraic multiplicity, respectively, of the eigenvalue �0 of F .

2. The sequence {'i,j}mj−1i=0 is a Jordan chain of F of length mj, for j = 1, . . . , g and

m1 ≥ ⋅ ⋅ ⋅ ≥ mg ≥ 1.

3. The vectors '0,j, j = 1, . . . , g form a basis for the eigenspace ker(F (�0)).

54

Example 3. Let A ∈Mn(ℂ) and define F (�) := A− �In for � ∈ ℂ. Then it can be shown

that {'i,j}mj−1i=0 , j = 1, . . . , g is a canonical system of Jordan chains of F corresponding to

the eigenvalue �0 if and only if the set of vectors∪gj=1{'i,j}

mj−1i=0 form a Jordan basis for the

generalized eigenspace of the matrix A corresponding to the eigenvalue �0 (see [35, p. 232,

§6.4] for the definition) where {'i,j}mj−1i=0 is a Jordan chain of A of length mj corresponding

to the eigenvalue �0 and m1 ≥ ⋅ ⋅ ⋅ ≥ mg ≥ 1.

Lemma 12 The relation ∼ defined by F ∼ G if F,G ∈ On×n(�0) are equivalent at �0, is

an equivalence relation on On×n(�0).

Proof. First, its obvious the relation ∼ is reflexive since the identity matrix In as a

constant function belongs in On×n(�0). Now lets show the relation is symmetric. Suppose

F,G ∈ On×n(�0) and F ∼ G. This implies there exists N,M ∈ On×n(�0) and an r > 0

such that F (�) = N(�)G(�)M(�) for ∣�−�0∣ < r and N(�0),M(�0) are invertible matrices.

It follows from [45, p. 7, §1.2, Proposition 1.2.5] that there exists a possibly smaller r > 0

such that N−1(�) := N(�)−1 and M−1(�) := M(�)−1 exist for ∣�− �0∣ < r and N−1,M−1 ∈

O(B(�0, r),Mn(ℂ)). But then N−1,M−1 ∈ On×n(�0), N−1(�),M−1(�0) are invertible, and

G(�) = N−1(�)F (�)M−1(�) for ∣�−�0∣ < r. This proves G ∼ F and so ∼ is symmetric. We

now prove∼ is transitive. Indeed, let F ∼ G and G ∼ H. Then there exists N1, N2,M1,M2 ∈

On×n(�0) and an r > 0 such that F (�) = N1(�)G(�)M1(�) = N1(�)N2(�)H(�)M1(�)M2(�)

for ∣�− �0∣ < r but the products N1N2 and M1M2 are in On×n(�0) and are invertible at �0.

Thus F ∼ H and so ∼ is transitive. This proves the relation ∼ is an equivalence relation on

On×n(�0).

Proposition 13 Let F,G be any holomorphic matrix function such that F,G ∈ On×n(�0)

are equivalent at �0. Then

1. �0 is an eigenvalue of F of finite algebraic multiplicity if and only if �0 is an eigenvalue

55

of G of finite algebraic multiplicity.

2. Let �0 be an eigenvalue of F of finite algebraic multiplicity. Then the local Smith

forms of F and G corresponding to the eigenvalue �0 are equal and, in particular, the

geometric, partial, and algebraic multiplicities of the eigenvalue �0 of F and G are the

same.

3. Let �0 be an eigenvalue of F of finite algebraic multiplicity. Let N,M ∈ On×n(�0) be

such that N(�0),M(�0) are invertible matrices and

F (�) = N(�)G(�)M(�) (3.15)

for every ∣�− �0∣ < r. If {'j}lj=0 is a Jordan chain of F of length l corresponding to

the eigenvalue �0 then {'j}lj=0 is a Jordan chain of G of length l corresponding to the

eigenvalue �0 where

'j :=1

j!

j∑ℎ=0

1

ℎ!M (ℎ)(�0)'j−ℎ, j = 0, . . . , l − 1. (3.16)

Proof. Let F,G be any holomorphic matrix function such that F,G ∈ On×n(�0) are

equivalent at �0. It follows from Lemma 12 and Theorem 9 that the local Smith forms of

F and G are equal. This implies that �0 is an eigenvalue of F finite algebraic multiplicity

if and only if �0 is an eigenvalue of G finite algebraic multiplicity. It also implies that if �0

is an eigenvalue of F finite algebraic multiplicity then the geometric, partial, and algebraic

multiplicities of the eigenvalue �0 of F and G are the same.

Suppose that �0 is an eigenvalue of F finite algebraic multiplicity. Let N,M ∈ On×n(�0) be

such that N(�0),M(�0) are invertible matrices and

F (�) = N(�)G(�)M(�) (3.17)

56

for every ∣�− �0∣ < r. Suppose {'j}lj=0 is a Jordan chain of F of length l corresponding to

the eigenvalue �0. Define the vector-valued function ' ∈ On×1(�0) by

'(�) :=l−1∑j=0

'j(�− �0)j, � ∈ ℂ.

It follows from Proposition 11 that ' is a generating function of order l for F at the eigenvalue

�0. This implies

'(�0) ∕= 0, F (�)'(�) = O((�− �0)l), as �→ �0.

But since N ∈ On×n(�0) and N(�0) is invertible this implies there exists a possibly smaller

r > 0 such that N−1(�) := N(�)−1 exists for ∣� − �0∣ < r and N−1 ∈ O(B(�0, r),Mn(ℂ)).

This implies

G(�)M(�)'(�) = N−1(�)F (�)'(�) = O((�− �0)l), as �→ �0.

But since M ∈ On×n(�0) and M(�0) is invertible then M' ∈ On×1(�0) and

M(�0)'(�0) ∕= 0, G(�)M(�)'(�) = O((�− �0)l), as �→ �0.

This implies by Definition 9 thatM' is a generating function of order l for F at the eigenvalue

�0 and so by Proposition 11, {'j}lj=0 is a Jordan chain of G of length l corresponding to the

eigenvalue �0 where

'j :=1

j!

dj

d�j(M(�)'(�))

∣∣�=�0

=1

j!

j∑ℎ=0

1

ℎ!M (ℎ)(�0)'j−ℎ, j = 0, . . . , l − 1.

This completes the proof.

Theorem 14 Let F ∈ On×n(�0) such that �0 an eigenvalue of F of finite algebraic mul-

57

tiplicity. Then there exists a canonical system of Jordan chains of F corresponding to the

eigenvalue �0.

Proof. Let F ∈ On×n(�0) such that �0 an eigenvalue of F of finite algebraic multiplicity.

Denote by g,m1, . . . ,mg,m the geometric, partial, and algebraic multiplicities, respectively,

of the eigenvalue �0 of F where m1 ≥ ⋅ ⋅ ⋅ ≥ mg ≥ 1 and m = m1 + ⋅ ⋅ ⋅ + mg. By Theorem

9 and Lemma 12 there exists N,M ∈ On×n(�0) and an r > 0 such that N(�0),M(�0) are

invertible matrices and

G(�) = N(�)F (�)M(�),

for every � ∈ B(�0, r) where

G(�) = diag{(�− �0)m1 , . . . , (�− �0)mg , 1, . . . , 1}.

Let e1, . . . , en denote the standard orthonormal basis vectors in ℂn. Define vector-valued

functions 'j ∈ On×1(�0), j = 1, . . . , g by

'j(�) = ej, � ∈ ℂ, j = 1, . . . , g. (3.18)

Then since

'j(�0) = ej ∕= 0, G(�)'i(�) = (�− �0)mjej = O((�− �0)mi), as �→ �0, (3.19)

each 'j is a generating function of order mj for G at the eigenvalue �0 and so by Proposition

11 the sequence {'i,j}mj−1i=0 defined by

'i,j :=1

i!'

(i)j (�0) = �0,iej, i = 0, . . . ,mj − 1, (3.20)

58

where �0,i is the Kronecker delta symbol, is a Jordan chain of G of length mj corresponding to

the eigenvalue �0, for j = 1, . . . , g. It follows that {'i,j}mj−1i=0 , j = 1, . . . , g is canonical system

of Jordan chains for G corresponding to the eigenvalue �0, that is, we have constructed

a canonical system of Jordan chains for the local Smith form of F corresponding to the

eigenvalue �0. By Proposition 13 the sequence {'i,j}mj−1i=0 defined by

'i,j :=1

i!

i∑ℎ=0

1

ℎ!M (ℎ)(�0)'i−ℎ,j, i = 0, . . . ,mj − 1 (3.21)

is a Jordan chain of F of length mj corresponding to the eigenvalue �0, for j = 1, . . . , g. More-

over, since e1, . . . , eg is a basis for kerG(�0), N(�0),M(�0) are invertible, and N(�0)G(�0) =

F (�0)M(�0) then '0,j = M(�0)ej, j = 1, . . . , g is a basis for kerF (�0). Thus we have shown

the numbers g, m1, . . . ,mg, and m = m1 + ⋅ ⋅ ⋅ + mg are the geometric multiplicity, partial

multiplicities, and algebraic multiplicity, respectively, of the eigenvalue �0 of F , the sequence

{'i,j}mj−1i=0 is a Jordan chain of F of length mj, for j = 1, . . . , g and m1 ≥ ⋅ ⋅ ⋅ ≥ mg ≥ 1, and

the vectors '0,j, j = 1, . . . , g form a basis for the eigenspace ker(F (�0)). Therefore, by Defi-

nition 10, {'i,j}mj−1i=0 , j = 1, . . . , g is a canonical system of Jordan chains of F corresponding

to the eigenvalue �0. This completes the proof.

It will be important in this thesis to know when an eigenvalue of a holomorphic matrix

function is both of finite algebraic multiplicity and semisimple. The next proposition gives

us a simple sufficient condition for this to be true.

Proposition 15 If �0 is an eigenvalue of F ∈ On×n(�0) and the equation

F (1)(�0)'0 = −F (�0)'1 (3.22)

has no solution for '0 ∈ ker(F (�0))/{0} and '1 ∈ ℂn then �0 is an eigenvalue of F of finite

algebraic multiplicity and a semisimple eigenvalue of F .

59

Proof. Let the hypotheses of this proposition be true for some F ∈ On×n(�0). Then the

hypothesis that (3.22) has no solution for '0 ∈ ker(F (�0))/{0} and '1 ∈ ℂn is equivalent

by Definition 8 to there are no Jordan chains of F of length l ≥ 2 corresponding to the

eigenvalue �0. Thus if �0 was an eigenvalue of F but not of finite algebraic multiplicity then

by Theorem 9 we could find a vector-valued function ' ∈ On×1(�0) and an r > 0 such that

F (�)'(�) = 0 for ∣� − �0∣ < r. By Proposition 11 we could find a Jordan chain of order l

for any l ∈ ℕ and in particular, for l ≥ 2, a contradiction. Hence �0 must be an eigenvalue

of F of finite algebraic multiplicity. If �0 was an eigenvalue of F but was not a semisimple

eigenvalue of F then by Theorem 14 there would exist a Jordan chain of F of length ≥ 2,

a contradiction. Therefore, �0 is an eigenvalue of F of finite algebraic multiplicity and a

semisimple eigenvalue of F . This completes the proof.

Another important result needed in this thesis is given by the next proposition.

Proposition 16 Let F ∈ On×n(�0) and let f ∈ On×n(z0) be such that f(z0) = �0. Define

G(z) := F (f(z)) for all z such that the composition is well-defined. Then G ∈ On×n(z0)

and �0 is an eigenvalue of F if and only if z0 is an eigenvalue of G. Moreover, if �0 is an

eigenvalue of F and f ′(z0) ∕= 0 then

1. �0 is an eigenvalue of F of finite algebraic multiplicity if and only if z0 is an eigenvalue

of G of finite algebraic multiplicity.

2. Let �0 be an eigenvalue of F of finite algebraic multiplicity. Then the geometric, partial,

and algebraic multiplicities of the eigenvalue �0 of F and of the eigenvalue z0 of G are

in one-to-one correspondence.

Proof. Let F ∈ On×n(�0) and let f ∈ On×n(z0) such that f(z0) = �0. It follows there

exists an � > 0 such that F ∈ O(B(�0, �),Mn(ℂ)). Thus there exists a � > 0 such that

f ∈ O(B(�0, �),ℂ)) and if z ∈ B(�0, �) then f(z) ∈ B(�0, �). It then follows G(z) = F (f(z))

60

is well-defined on B(�0, �) and G′(z) = f ′(z)F ′(f(z)) for every z ∈ B(�0, �). Thus G ∈

O(B(z0, �),Mn(ℂ)) and so G ∈ On×n(z0). It follows from the fact detF (�0) = detG(z0)

that �0 is an eigenvalue of F if and only if z0 is an eigenvalue of G.

Suppose �0 is an eigenvalue of F . Then by the local Smith form, Theorem 9, there exists

N,M ∈ On×n(�0) and an r > 0 with r ≤ � such that N(�0),M(�0) are invertible matrices

and

F (�) = N(�) diag{0�, (�− �0)m�+1 , . . . , (�− �0)mg , In−g}M(�),

for every � ∈ B(�0, r), where 0 ≤ � ≤ g ≤ n, 0� is the � × � zero matrix, In−g is the

(n−g)×(n−g) identity matrix, the number of zeros down the diagonal is �, when � = g there

are no (�− �0) terms appearing in the diagonal, when � < g we have m�+1 ≥ . . . ≥ mg ≥ 1

with {m�+j}gj=1 ⊆ ℕ, and when g = n there are no ones appearing in the diagonal.

Again we can find a �1 ≤ � such that f(z) ∈ B(�0, r) for z ∈ B(z0, �1). But from what we

just proved this implies for N(z) := N(f(z)) and M(z) := M(f(z)), for all z such that the

composition is well-defined, that N , M ∈ On×n(z0) with N(z0) = N(�0), M(z0) = M(�0)

invertible matrices such that

G(z) = F (f(z))

= N(z) diag{0�, (f(z)− �0)m�+1 , . . . , (f(z)− �0)mg , In−g}M(z)

for every z ∈ B(z0, �1).

Now since f ′(z0) ∕= 0 then for the function f0(z) := f(z)z−z0 for z ∈ B(z0, �1) we have

O(B(z0, �1),ℂ) with f0(z0) = f ′(z0) ∕= 0. Thus there exists a �2 > 0 with �2 ≤ �1 such

that f−10 (z) := z−z0

f(z)with z ∈ B(z0, �2) is well-defined and f−1

0 ∈ O(B(z0, �2),ℂ).

61

Define the matrix M(z) for z ∈ B(z0, �2) by

M(z) = diag{I�, f0(z)m�+1 , . . . , f0(z)mg , In−g}M(z).

It follows from this that M ∈ O(B(z0, �2),Mn(ℂ)) with M(z0) an invertible matrix and M(z)

is invertible for every z ∈ B(z0, �2). Moreover, since we have (f(z)− �0)j = (z − z0)jf0(z)j,

it follows that

G(z) = F (f(z))

= N(z) diag{0�, (f(z)− �0)m�+1 , . . . , (f(z)− �0)mg , In−g}M(z)

= N(z) diag{0�, (z − z0)m�+1 , . . . , (z − z0)mg , In−g}M(z)

for every z ∈ B(z0, �2).

Therefore by the uniqueness portion of Theorem 9 it follows that the local Smith form of F

corresponding to the eigenvalue �0 is

diag{0�, (�− �0)m�+1 , . . . , (�− �0)mg , In−g}

and the local Smith form of G corresponding to the eigenvalue z0 is

diag{0�, (z − z0)m�+1 , . . . , (z − z0)mg , In−g}.

The proof of the proposition now follows from the uniqueness of those local Smith forms.

The following example arises in our thesis in Floquet theory when studying the Jordan normal

form of the monodromy matrix for canonical systems of periodic differential equations with

period d.

62

Example 4. Let A ∈ Mn(ℂ) and define F (�) := A − �In for � ∈ Mn(ℂ). Define G(k) :=

F (eikd) = A−eikdIn, for k ∈ ℂ. Let �0 be an eigenvalue of the matrix A. Then we know from

example 1 that if we let g,m1 ≥ ⋅ ⋅ ⋅ ≥ mg,m := m1 + ⋅ ⋅ ⋅+mg denote the geometric, partial,

and algebraic multiplicities, respectively, of the eigenvalue �0 of F then the Jordan normal

form of the matrix A corresponding to the eigenvalue �0 has exactly g Jordan blocks of order

m1, . . . ,mg, respectively, with eigenvalue �0. Now let k0 ∈ ℂ such that eik0d = �0. Then,

since ddkeikd = ideikd ∕= 0 for any k, it follows from Proposition 16 that k0 is an eigenvalue of G

with finite algebraic multiplicity such that its geometric, partial, and algebraic multiplicities

are g, m1, . . . ,mg and m, respectively. By the same proposition the converse is also true.

We conclude by noting that it can be helpful sometimes to study the spectrum of the matrix

A by studying the spectrum of the holomorphic matrix function G(k) := A − eikdIn. This

perspective is found useful when considering perturbation problems such as when A := A(!)

depends on a perturbation parameter !.

3.2 On Holomorphic Matrix-Valued Functions of Two

Variables

We introduce the following notation and convention for this section:

(i) O(U,E) := {f : U → E) ∣ f is holomorphic on U} where U is an open connected set

in ℂ2, (E, ∣∣ ⋅ ∣∣) is a Banach space, and a function f : U → E is said to be holomorphic

on U provided the partial derivatives of f exist at each point in U , i.e., the limits

∂f

∂�

∣∣∣(�,z)=(�0,z0)

:= f�(�0, z0) := lim�→�0

(�− �0)−1(f(�, z0)− f(�0, z0))

∂f

∂z

∣∣∣(�,z)=(�0,z0)

:= fz(�0, z0) := limz→z0

(z − z0)−1(f(�0, z)− f(�0, z0))

63

exist for every (�0, z0) ∈ U .

(ii) O((�0, z0)) :=∪r>0

O(B((�0, z0), r),ℂ) – the set of scalar functions holomorphic at

(�0, z0) ∈ ℂ2.

(iii) Om×n((�0, z0)) :=∪r>0

O(B((�0, z0), r),Mm,n(ℂ)) – the set of m×n matrix-valued func-

tions holomorphic at (�0, z0) ∈ ℂ2.

(iv) As convention we identify O1×1((�0, z0)) with O((�0, z0)).

(v) If f1, f2 ∈ Om×n((�0, z0)) then we use the notation f1 = f2 to mean there exists

an r > 0 such that f1, f2 ∈ O(B((�0, z0), r),Mm,n(ℂ)) and f1(�, z) = f2(�, z) for

∣∣(�, z)− (�0, z0)∣∣ℂ < r.

The following is an important characterization of holomorphic functions as analytic functions:

Lemma 17 If f ∈ Om×n((�0, z0)) then f(�, z) is analytic at (�, z) = (�0, z0), that is,

the partial and mixed partial derivatives f (j1,j2)(�0, z0) := ∂j1+j2f∂j1�∂j2z

∣∣(�,z)=(�0,z0)

exist for ev-

ery j1, j2 ∈ ℕ ∪ {0} and there exists an r > 0 such that

f(�, z) =∞∑

j1,j2=0

1

j1!j2!f (j1,j2)(�0, z0)(�− �0)j1(z − z0)j2 , ∣�− �0∣ < r, ∣z − z0∣ < r

where the power series on the right converges absolutely in the Mm,n(ℂ) norm to f(�, z) for

(�, z) ∈ B(�0, r)×B(z0, r).

Proof. The space Mm,n(ℂ) is finite dimensional and since all norms on a finite dimensional

space are equivalent, we need only show that the statement is true for the norm given by

∣∣[ai,j]m,ni=1,j=1∣∣∞ := maxi,j∣ai,j∣, [ai,j]

m,ni=1,j=1 ∈Mm,n(ℂ).

64

Thus it follows from this that Om×n((�0, z0)) = Mm,n(O((�0, z0))) and the statement will be

true if we can show that its true for every f ∈ O((�0, z0)). But this statement for functions

in O((�0, z0)) is a well known fact from the theory of functions of several complex variables

[17, p. 39, Theorem 3.2] and follows from a deep theorem of Hartogs (see [17, p. 21, Theorem

1.6]).

In the spectral perturbation theory of holomorphic matrix functions, the following theorem

plays a central role:

Theorem 18 (Weierstrass Preparation Theorem) If f ∈ O((�0, z0)), f(�0, z0) = 0,

and f ∕= 0, then there exists an r > 0 such that

f(�, z) = (z − z0)�f0(�, z)f1(�, z), f1(�, z) ∕= 0, and (3.23)

f0(�, z) = (�− �0)m + a1(z)(�− �0)m−1 + ⋅ ⋅ ⋅+ am−1(z)(�− �0) + am(z) (3.24)

for every (�, z) ∈ B(�0, r) × B(z0, r). Here �,m ∈ ℕ ∪ {0}, aj ∈ O(B(z0, r),ℂ) with

aj(z0) = 0 for j = 1, . . . ,m− 1, and f1 ∈ O(B(�0, r)×B(z0, r),ℂ).

Proof. The proof of this statement follows from [17, p. 55, §I.4, Theorem 4.1].

It follows from this that to study the zero set of the function f ∈ O((�0, z0)) near (�0, z0)

we need to study the zero set of the monic polynomial f0 in � (if m ≥ 1) given by (3.24),

whose coefficients a1, . . . , am are in O(z0) and satisfy a1(z0) = ⋅ ⋅ ⋅ = am(z0) = 0.

Hence we define the set

O0((�0, z0)) := {f ∈ O((�0, z0)) ∣ f has the form in (3.24)}. (3.25)

65

Let f ∈ O0((�0, z0)). Then f(�, z0) = (�− �0)m for some m ∈ ℕ ∪ {0}. We call deg f := m

the degree of f .

A function f ∈ O0((�0, z0)) with f ∕= 1 is said to be irreducible if whenever f = p1p2 with

p1, p2 ∈ O0((�0, z0)) then p1 = 1 or p2 = 1.

Theorem 19 Every f ∈ O0((�0, z0)) with f ∕= 1, factors uniquely into a finite product of

irreducibles, i.e., there exists unique irreducibles p1, . . . , pl ∈ O0((�0, z0)) (not necessarily

distinct) such that

f =l∏

j=1

pj. (3.26)

Furthermore, deg f = deg p1 + ⋅ ⋅ ⋅ + deg pl. Moreover, there exists an r > 0 such that for

each j = 1, . . . , l and for each fixed z satisfying 0 < ∣z − z0∣ < r, the polynomial pj(�, z) has

only simple roots.

Proof. The uniqueness of a factorization into irreducibles follows from [4, p. 397, §A.2.6,

Theorem 1] and our choice of notation in A.2.(v). From the definition of the degree of an

element from O0((�0, z0)) we have

(�− �0)deg f = f(�, z0) =l∏

j=1

pj(�, z0) =l∏

j=1

(�− �0)deg pj = (�− �0)deg p1+⋅⋅⋅+deg pl

implying deg f = deg p1 + ⋅ ⋅ ⋅ + deg pl. The latter part of this theorem follows from [4, p.

399, §A.3.2, Theorem 2].

Definition 11 Let q ∈ ℕ and r > 0. If v ∈ O(B(0, q√r),Mm,n(ℂ)), where q

√r denotes the

real qth root of r, and is given by the convergent power series

v(") =∞∑j=0

cj"j, ∣"∣ < q

√r (3.27)

66

then the multi-valued function

u(z) := v((z − z0)1q ) =

∞∑j=0

cj(z − z0)jq , ∣z − z0∣ < r, (3.28)

is called a convergent Puiseux series expanded about z0 with limit point c0, domain

B(z0, r), and period q.

Fix any branch of the qth root function and denote its evaluation at (z − z0) by (z − z0)1/q

(e.g., using the principal branch of the qth root (z − z0)1/q = ∣z − z0∣1/qe1qiarg(z−z0) where

−� < arg(z−z0) ≤ �). Then the single-valued functions u0, . . . , uq−1 given by the convergent

series

uℎ(z) := v(�ℎ(z − z0)1/q) =∞∑j=0

cj(�ℎ(z − z0)1/q)j, ∣z − z0∣ < r, ℎ = 0, . . . , q − 1, (3.29)

where � is any primitive qth root of unity (e.g., � = ei2�/q), are called the branches of the

Puiseux series u. The values of the Puiseux series u at a point z ∈ B(z0, r) are the values

of its branches at this point, namely, u0(z), . . . , uq−1(z). The graph of the Puiseux series u

is the union of the graphs of its branches, namely,∪q−1ℎ=0{(z, uℎ(z)) : ∣z − z0∣ < r}.

Theorem 20 Let f ∈ O0((�0, z0)) with f ∕= 1. Let f =l∏

j=1

pj be a factorization of f into

unique irreducibles p1, . . . , pl ∈ O0((�0, z0)) (not necessarily distinct) with deg f = deg p1 +

⋅ ⋅ ⋅ + deg pl and r > 0 be such that for each j = 1, . . . , l and for each fixed z satisfying

0 < ∣z−z0∣ < r, the polynomial pj(�, z) has only simple roots. Then there exists l convergent

Puiseux series �1(z), . . . , �l(z) expanded about z0 with limit point �0, domain B(z0, r), and

periods deg p1, . . . , deg pl, respectively, such that for every (�, z) ∈ B(z0, r)× ℂ,

pj(�, z) =

deg pj−1∏ℎ=0

(�− �j,ℎ(z)), j = 1, . . . , l (3.30)

67

and

f(�, z) =l∏

j=1

deg pj−1∏ℎ=0

(�− �j,ℎ(z)), (3.31)

where �j,ℎ(z), ℎ = 0, . . . , deg pj − 1 denotes all the branches of the Puiseux series �j(z), for

j = 1, . . . , l.

Proof. This statement follows from [4, p. 406, §A.5.4, Theorem 3].

3.3 On the Perturbation Theory for Holomorphic Ma-

trix Functions

In this section, we consider a holomorphic matrix-valued function of two variables L ∈

On×n((�0, z0)) which satisfies det(L(�0, z0)) = 0. For such a function, we can consider

L(�, z) as a family, indexed by a perturbation parameter z, of holomorphic matrix functions

in the variable �. We will give basic facts regarding the dependency on z of the eigenvalues

�(z) and corresponding eigenvectors '(z) of the holomorphic matrix function L(�, z) when

(�, z) is near (�0, z0). In order to avoid confusion we denote by L(⋅, z), the holomorphic

matrix function L(�, z) in the variable � for a fix z.

3.3.1 Eigenvalue Perturbations

We begin by defining the spectrum for a holomorphic matrix-valued function of two variables

in a manner similar to the one variable case.

Definition 12 If L ∈ O(U,Mn(ℂ)) then the spectrum of L, denoted by �(L), is the subset

68

of U such that the matrix L(�, z) is not invertible, i.e.,

�(L) := {(�, z) ∈ U ∣ detL(�, z) = 0}. (3.32)

Notice that (�0, z0) ∈ �(L) if and only if �0 ∈ �(L(⋅, z0)), that is, �0 is an eigenvalue of the

holomorphic matrix function L(⋅, z0) ∈ O(�0). Thus the spectral theory developed in the

previous sections for holomorphic matrix functions applies.

Conceptually, we can consider the eigenvalues �(z) of L(⋅, z) as a multi-valued function of z

implicitly defined by the equation detL(�, z) = 0. Thus it is useful to introduce the notation

�(L)−1 for its graph, i.e.,

�(L)−1 := {(z, �) ∣ (�, z) ∈ �(L)}. (3.33)

With this in mind, the next theorem can be interpreted as a description of the local analytic

properties of this multi-valued function.

Theorem 21 Suppose L ∈ O(U,Mn(ℂ)) such that (�0, z0) ∈ �(L). If �0 an eigenvalue

of L(⋅, z0) with algebraic multiplicity m then for any " > 0 there exists an � > 0 such

that �(L)−1 ∩ B(z0, �) × B(�0, ") is the union of the graphs of l convergent Puiseux series

�1(z), . . . , �l(z) expanded about z0 with limit point �0, domain B(z0, �), and periods q1, . . . , ql,

respectively, which satisfy m = q1 + ⋅ ⋅ ⋅+ ql.

Proof. First, L ∈ O(U,Mn(ℂ)) implies f := det(L) ∈ O(U,ℂ) since the determinant of

a matrix is just a polynomial in its entries. Hence �(L) is the zero set of the holomorphic

function f : U → ℂ and since (�0, z0) ∈ �(L) then f(�0, z0) = 0. Suppose �0 an eigen-

value of L(⋅, z0) with algebraic multiplicity m. Then by Proposition 10 the order of the

zero of f(�, z0) = det(L(�, z0)) at � = �0 is m and so f(⋅, z0) ∕= 0. By the Weierstrass

69

Preparation Theorem, Theorem 18 above, there exists f0 ∈ O0((�0, z0)) with degree m and

f1 ∈ O((�0, z0)) with f1(�0, z0) ∕= 0 such that f = f0f1 in O((�0, z0)). By Theorem 19,

there exists unique irreducibles p1, . . . , pl ∈ O0((�0, z0)) (not necessarily distinct) such that

in O((�0, z0)) we have

f = f1f0 = f1

l∏j=1

pj.

Moreover, by the same theorem, deg f0 = deg p1 + ⋅ ⋅ ⋅+ deg pl and there exists an r > 0 such

that for each j = 1, . . . , l and for each fixed z satisfying 0 < ∣z − z0∣ < r, the polynomial

pj(�, z) has only simple roots. By taking a smaller r, we may also assume without loss of

generality that f, f1, p1, . . . , pl ∈ O(B(�0, r)×B(z0, r),ℂ) and

f(�, z) = f1(�, z)l∏

j=1

pj(�, z), f1(�, z) ∕= 0,

for every (�, z) ∈ B(�0, r)× B(z0, r). We now define the numbers qj := deg pj, j = 1, . . . , l.

Thus m = deg f0 = deg p1 + ⋅ ⋅ ⋅ + deg pl = q1 + ⋅ ⋅ ⋅ + ql. By Theorem 20 there exists l

convergent Puiseux series �1(z), . . . , �l(z) expanded about z0 with limit point �0, domain

B(z0, r), and periods q1, . . . , ql, respectively, such that

pj(�, z) =

qj−1∏ℎ=0

(�− �j,ℎ(z)), j = 1, . . . , l

where �j,ℎ(z), ℎ = 0, . . . , qj − 1 denotes all the branches of the Puiseux series �j(z), for

j = 1, . . . , l. This implies

det(L(�, z)) = f1(�, z)l∏

j=1

qj−1∏ℎ=0

(�− �j,ℎ(z)), f1(�, z) ∕= 0,

for every (�, z) ∈ B(�0, r)×B(z0, r).

70

Let " > 0 be given. Then since

limz→z0

�j,ℎ(z) = �j,ℎ(z0) = �0, ℎ = 0, . . . , qj − 1, j = 1, . . . , l,

there exists � > 0 with � ≤ r such that

∣�j,ℎ(z)− �0∣ < min{", r}, ℎ = 0, . . . , qj − 1, j = 1, . . . , l,

if ∣z − z0∣ < �. Then, since B(�0, ")×B(z0, �) ⊆ B(�0, r)×B(z0, r), we have

�(L) ∩B(�0, ")×B(z0, �) = �(L) ∩B(�0, r)×B(z0, r) ∩B(�0, ")×B(z0, �)

=l∪

j=1

qj−1∪ℎ=0

{(�j,ℎ(z), z) : (�j,ℎ(z), z) ∈ B(�0, r)×B(z0, r)} ∩B(�0, ")×B(z0, �)

=l∪

j=1

qj−1∪ℎ=0

{(�j,ℎ(z), z) ∣ z ∈ B(z0, �) }

It follows from this that if we just take the convergent Puiseux series �1(z), . . . , �l(z) to have

the domain B(z0, �) instead of B(z0, r) then we have l convergent Puiseux series expanded

about z0 with limit point �0, domain B(z0, �), and periods q1, . . . , ql, respectively, which

satisfy m = deg q1 + ⋅ ⋅ ⋅+ deg ql and such that the union of their graphs is

l∪j=1

qj−1∪ℎ=0

{(z, �j,ℎ(z)) ∣ z ∈ B(z0, �) } = {(z, �) ∣ (�, z) ∈ �(L) ∩B(�0, ")×B(z0, �)}

= �(L)−1 ∩B(z0, �)×B(�0, ").

This completes the proof.

Let L ∈ O(U,Mn(ℂ)) and assume that (�0, z0) ∈ �(L) and �0 an eigenvalue of L(⋅, z0) with

finite algebraic multiplicity. Denote by g, m1 ≥ . . . ≥ mg, and m = m1 + ⋅ ⋅ ⋅ + mg the

geometric multiplicity, partial multiplicities, and algebraic multiplicity, respectively of the

71

eigenvalue �0 of L(⋅, z0). Then it follows Theorem 21 that given an " > 0 there exists a � > 0

such that, for ∣z−z0∣ < �, the spectrum of L(�, z) in ∣�−�0∣ < ", that is, �(L(⋅, z))∩B(�0, "),

consists of m eigenvalues and they are the values of l convergent Puiseux series of the form

�j(z) = �0 +∞∑s=1

cj,s(z − z0)sqj , ∣z − z0∣ < �, j = 1, . . . , l (3.34)

where there periods satisfy m = q1 + . . . + ql. Specifically, letting �j,ℎ(z), ℎ = 0, . . . , qj − 1

denote all the branches of the Puiseux series �j(z), for j = 1, . . . , l then

�(L(⋅, z)) ∩B(�0, ") =l∪

j=1

{�j,ℎ(z) : ℎ = 0, . . . , qj − 1} (3.35)

for each z ∈ B(z0, �).

In general the only connection between the numbers l and q1, . . . , ql with the geometric mul-

tiplicity g and partial multiplicities m1, . . . ,mg is the equality m =∑l

j=1 qj =∑g

j=1mj.

Because of this two important problems arise in the spectral perturbation theory of holo-

morphic matrix functions:

1. Find general conditions involving just the partial derivatives of L at (�0, z0) up to first

order, namely, L(�0, z0), L�(�0, z0), and Lz(�0, z0) such that we may determine the

values l, q1, . . . , ql and, in particular, those conditions which imply l = g and qj = mj,

for j = 1, . . . , g.

2. Find formulas for the first order coefficients of those Puiseux series, i.e., formulas for

coefficient cj,1, j = 1, . . . , l in (3.34).

Several papers have addressed these problems (see [23, 34, 37–39]) and significant progress

was made towards their resolution.

72

3.3.2 Eigenvector Perturbations

We will now proceed to give a useful description for the dependency on z of the perturbed

eigenvectors '(z) of the holomorphic matrix function L(�, z) corresponding to perturbed

eigenvalue �(z) when (�, z) is near a point (�0, z0) ∈ �(L).

Definition 13 Let L ∈ O(U,Mn(ℂ)). We say �(z) is a eigenvalue Puiseux series of

L(⋅, z) provided it is a convergent Puiseux series whose values at each point z in its domain

are eigenvalues of L(⋅, z). If

�(z) = �0 +∞∑s=1

cs(z − z0)sq , ∣z − z0∣ < � (3.36)

is a eigenvalue Puiseux series of L(⋅, z) then we say '(z) is an eigenvector Puiseux series

of L(⋅, z) corresponding to the eigenvalue Puiseux series �(z) provided it is a convergent

Puiseux series with

'(z) = �0 +∞∑s=1

�s(z − z0)sq , ∣z − z0∣ < r (3.37)

where 0 < r ≤ �, {�s}∞s=0 ⊆ ℂn, �0 ∕= 0, and such that if we fix any branch of the qth root

function and denote its evaluation at (z − z0) by (z − z0)1/q then their branches

�ℎ(z) = �0 +∞∑s=1

cs(�ℎ(z − z0)1/q)s, ∣z − z0∣ < r, ℎ = 0, . . . , q, (3.38)

'ℎ(z) = �0 +∞∑s=1

�s(�ℎ(z − z0)1/q)s, ∣z − z0∣ < r, ℎ = 0, . . . , q, (3.39)

where � is any primitive qth root of unity, satisfy

L(�ℎ(z), z)'ℎ(z) = 0, 'ℎ(z) ∕= 0, ℎ = 0, . . . , q − 1, ∣z − z0∣ < r. (3.40)

73

We call the pair (�(z), '(z)) an eigenpair Puiseux series of L(⋅, z).

The following lemma tells us that to every eigenvalue Puiseux series there exists a corre-

sponding eigenvector Puiseux series.

Lemma 22 Suppose L ∈ O(U,Mn(ℂ)). If �(z) is a eigenvalue Puiseux series of L(⋅, z) then

there exists an eigenvector Puiseux series '(z) of L(⋅, z) corresponding to the eigenvector

Puiseux series �(z).

Proof. Assume �(z) be an eigenvalue Puiseux series of L(⋅, z). This implies for some

q ∈ ℕ, � > 0, v ∈ O(B(0, q√�),ℂ), and some (�0, z0) ∈ U , �(z) is a convergent Puiseux series

with

�(z) = v((z − z0)1q ) = �0 +

∞∑s=1

cs(z − z0)sq , ∣z − z0∣ < � (3.41)

and taking any fixed branch of the qth root and denoting its evaluation at z−z0 by (z−z0)1/q,

its branches

�ℎ(z) = v(�ℎ(z − z0)1/q) = �0 +∞∑s=1

cs(�ℎ(z − z0)1/q)s, ∣z − z0∣ < r, ℎ = 0, . . . , q, (3.42)

where � is any primitive qth root of unity, satisfy (�ℎ(z), z) ∈ U for ∣z − z0∣ < � and

detL(�ℎ(z), z) = 0, ∣z − z0∣ < �, ℎ = 0, . . . , q − 1, (3.43)

since the values �ℎ(z), ℎ = 0, . . . , q−1 for ∣z−z0∣ < � are eigenvalues of L(⋅, z) by assumption.

We introduce a new variable " ∈ B(0, q√�). Then z(") := "q + z0 ∈ B(z0, �) is an analytic

function. Moreover, for each " ∈ B(0, q√�) there exists an ℎ ∈ {0, . . . , q − 1} such that " =

�ℎ(z(")−z0)1/q. This implies (v("), z(")) = (�ℎ(z(")), z(")) ∈ U and hence detL(v("), z(")) =

74

detL(�ℎ(z), z) = 0. Thus we have shown for every " ∈ B(0, q√r), the analytic functions v(")

and z(") satisfy (v("), z(")) ∈ U and so have ranges in the domain of the analytic function

L and satisfy

detL(v("), z(")) = 0. (3.44)

But this implies the function F : B(0, q√�) → Mn(ℂ) defined by F (") := L(v("), z(")) is an

analytic function as well, i.e., F ∈ O(B(0, q√�),Mn(ℂ)), and hence detF ∈ O(B(0, q

√�),ℂ)

with

detF = 0. (3.45)

But this implies that 0 is an eigenvalue of the holomorphic matrix function F of infinite

algebraic multiplicity. Consider the local Smith form G of F corresponding to the eigenvalue

0 of infinite algebraic multiplicity. By Theorem 9 it must be of the form

G(") = diag{0, . . . , 0, "m′�+1 , . . . , "m′g , 1, . . . , 1} (3.46)

where 1 ≤ � ≤ g = dim kerF (0) = dim kerL(�0, z0) ≤ n, the number of zeros down the

diagonal is 1 ≤ � = dim kerF (") = dim kerL(v("), z(")) for 0 < " ≪ 1, when � = g

there are no " terms down the diagonal, when � < g we have m′�+1 ≥ ⋅ ⋅ ⋅ ≥ m′g ≥ 1 with

{m′�+j}gj=1 ⊆ ℕ, and when g = n there are no ones appearing in the diagonal. Hence by

Theorem 9 and Lemma 12 there exists an r > 0 with 0 < q√r ≤ q√� and matrix functions

M,N ∈ O(B(0, q√r),Mn(ℂ)) such that M("), N(") are invertible matrices for every " ∈

B(0, q√r) and satisfy

diag{0, . . . , 0, "m′�+1 , . . . , "m′g , 1, . . . , 1} = N(")F (")M(")

= N(")L(v("), z("))M("), ∣"∣ < q√r. (3.47)

75

Denote the standard orthonormal basis vectors in ℂn by e1, . . . , en. Then

�j(") := M(")ej, j = 1, . . . , � (3.48)

are the first � columns of M(") and so are a basis for kerL(v("), z(")) for every " ∈ B(0, q√r).

Hence for j = 1, . . . , � we have �j ∈ O(B(0, q√r),ℂn) and �j(") ∕= 0 for every " ∈ B(0, q

√r).

In particular, �1 ∈ O(B(0, q√r),ℂn) and satisfies

L(v("), z("))'1(") = 0, '1(") ∕= 0, ∣"∣ < q√r. (3.49)

Now since �1 ∈ O(B(0, q√r),ℂn) it follows that it has a power series expansion about the

point z0 which converges for ∣"∣ < q√r. Thus we have

�1(z) = �0 +∞∑s=1

�s"s, ∣"∣ < q

√r. (3.50)

where {�s}∞s=0 ⊆ ℂn, �0 ∕= 0, and the series converges absolutely in the ℂn norm for every

" ∈ B(0, q√r). It follows that a convergent Puiseux series '(z) is given by

'(z) = �1((z − z0)1q ) = �0 +

∞∑s=1

�s(z − z0)sq , ∣z − z0∣ < r. (3.51)

We will now show that '(z) is an eigenvector Puiseux series of L(⋅, z) corresponding to

the eigenvalue Puiseux series �(z). Fix any branch of the qth root function denoting its

evaluation at (z − z0) by (z − z0)1/q and let � be any primitive qth root of unity. Now if

∣z − z0∣ < r then "ℎ(z) := �ℎ(z − z0)1/q ∈ B(0, q√r) ⊆ B(0, q

√�) for ℎ = 0, . . . , q and so it

follows that the branches of �(z) and '(z) are given by the convergent series

�ℎ(z) = �0 +∞∑s=1

cs(�ℎ(z − z0)1/q)s, ∣z − z0∣ < r, ℎ = 0, . . . , q, (3.52)

76

'ℎ(z) = �0 +∞∑s=1

�s(�ℎ(z − z0)1/q)s, ∣z − z0∣ < r, ℎ = 0, . . . , q. (3.53)

Moreover, if ∣z − z0∣ < r then we have "ℎ(z) = �ℎ(z − z0)1/q ∈ B(0, q√r) ⊆ B(0, q

√�) implies

z = z("ℎ(z)), �ℎ(z) = v("ℎ(z)), (v("ℎ(z)), z("ℎ(z))) = (�ℎ(z), z) ∈ U , 'ℎ(z) = �1("ℎ(z)) ∕= 0,

and

L(�ℎ(z), z)'ℎ(z) = L(v("ℎ(z)), z("ℎ(z)))�1("ℎ(z)) = 0 (3.54)

for ℎ = 0, . . . , q− 1. Therefore '(z) is an eigenvector Puiseux series of L(⋅, z) corresponding

to the eigenvalue Puiseux series �(z). This completes the proof.

Theorem 23 Suppose L ∈ O(U,Mn(ℂ)) such that (�0, z0) ∈ �(L). If �0 an eigenvalue of

L(⋅, z0) with algebraic multiplicity m then for any " > 0 there exists a r > 0 and l eigenvalue

Puiseux series (�1(z), '1(z)), . . . , (�l(z), 'l(z)) expanded about z0 with domain B(z0, r) and

periods q1, . . . , ql, respectively, satisfying m = q1 +⋅ ⋅ ⋅+ql such that if we fix any branch of the

qth root function, for q ∈ {q1, . . . , ql}, denoting its evaluation at (z − z0) by (z − z0)1/q and

let �q be any primitive qth root of unity then their branches given by the convergent series

�j,ℎ(z) = �0 +∞∑s=1

cj,s(�ℎqj

(z − z0)1/qj)s, ∣z − z0∣ < r, ℎ = 0, . . . , qj − 1, (3.55)

'j,ℎ(z) = �j,0 +∞∑s=1

�j,ℎ(�ℎ(z − z0)1/qj)s, ∣z − z0∣ < r, ℎ = 0, . . . , qj − 1. (3.56)

satisfy

detL(�j,ℎ(z), z) = 0, ℎ = 0, . . . , qj − 1, j = 1, . . . , l, ∣z − z0∣ < r (3.57)

�(L) ∩B(�0, ")×B(z0, r) =l∪

j=1

qj−1∪ℎ=1

{(�j,ℎ(z), z) : ∣z − z0∣ < r} (3.58)

�(L(⋅, z)) ∩B(�0, ") =l∪

j=1

qj−1∪ℎ=1

{�j,ℎ(z)}, ∣z − z0∣ < r (3.59)

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L(�ℎ(z), z)'ℎ(z) = 0, 'ℎ(z) ∕= 0, ℎ = 0, . . . , qj − 1, j = 1, . . . , l, ∣z − z0∣ < r. (3.60)

Proof. This theorem follows directly from Theorem 21 and Lemma 22.

Now in general the zero-order eigenvectors {�j,0}gj=1 ⊆ kerL(�0, z0) need not be a basis for

kerL(�0, z0), the eigenspace of L(⋅, z0) corresponding to the eigenvalue �0. Because of this

two important problems arise in the spectral perturbation theory of holomorphic matrix

functions:

1. Find general conditions involving just the partial derivatives of L at (�0, z0) up to first

order, namely, L(�0, z0), L�(�0, z0), and Lz(�0, z0) such that we may determine whether

or not we can choose eigenpair Puiseux series (�1(z), '1(z)), . . . , (�l(z), 'l(z)) of L(⋅, z)

as given in the above theorem so that their zero-order eigenvectors {'j(z0)}gj=1 span

the whole eigenspace kerL(�0, z0).

2. Find formulas to determine the zero-order eigenvectors {�j,0}gj=1.

The literature which addresses these problems is scarce and this author is only aware of

[23, 34] which addresses these problems for general perturbations of holomorphic matrix

functions. Even in the special case L(�, z) = A(z)− �I, few results exist with the exception

of [4, p. 270, Theorem 4; pp. 310-311, Theorem 8, Corollary 2, & Corollary 3], [41, Theorem

2], [58, p. 64, Chap. 2, §7.10, Theorem 6], and [47, Theorem 2.2]. In particular, the second

problem remains open even in this case, when total splitting at the first-order approximation

(see [4, p. 311] for definition) does not occur, and seems a difficult problem to resolve.

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3.3.3 Analytic Eigenvalues and Eigenvectors

In this section we give a survey of the main results from [23, 34] that we will need in our

thesis.

Preliminaries. We first begin with some preliminaries.

Definition 14 Let F ∈ O(U,Mm,n(ℂ)) where U ⊆ ℂ is an open connected set. Define

U∗ := {� ∣ � ∈ U}. The matrix function F ∗ : U∗ → Mm,n(ℂ) defined by F ∗(�) = F (�)∗ is

called the adjoint of the holomorphic matrix function F .

Lemma 24 If F ∈ O(U,Mm,n(ℂ)) where U ⊆ ℂ is an open connected set then U∗ is an

open connected set and F ∗ ∈ O(U∗,Mm,n(ℂ)).

Proof. If F ∈ O(U,Mm,n(ℂ)) then F = [aij]m,ni=1,j=1 where aij ∈ O(U,ℂ). Now obviously

U∗ is an open connected set and from complex analysis we know a∗ij(�) := aij(�) belongs to

O(U∗,ℂ). But this implies F ∗ = [a∗ji]n,mi=1,j=1 ∈ O(U∗,Mm,n(ℂ)). This completes the proof.

Proposition 25 Let F ∈ O(U,Mn(ℂ)). Then

1. �0 ∈ ℂ is an eigenvalue of F if and only if �0 is an eigenvalue of F ∗.

2. �0 is an eigenvalue of F of finite algebraic multiplicity if and only if �0 is an eigenvalue

of F ∗ of finite algebraic multiplicity.

3. If �0 is an eigenvalue of F of finite algebraic multiplicity then the geometric, partial,

and algebraic multiplicities of �0 coincide with the geometric, partial, and algebraic

multiplicities, respectively, of the eigenvalue �0 of F ∗.

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Proof. The first and second properties follows from the equality detF (�0) = detF (�0)∗ =

detF ∗(�0) for any �0 ∈ U . To prove the third part let �0 be an eigenvalue of F of finite

algebraic multiplicity and let g,m1 ≥ ⋅ ⋅ ⋅ ≥ mg,m := m1 + . . .+mg be its geometric, partial,

and algebraic multiplicities, respectively. Then by Theorem 9 there exists N,M ∈ O(�0)

and an r > 0 such that M(�0), N(�0) are invertible and for every � ∈ B(�0, r) we have

F (�) = N(�) diag{(�− �0)m1 , . . . , (�− �0)mg , 1, . . . , 1}M(�).

where if g = n there are no ones appearing in the diagonal. But this implies for any

� ∈ B(�0, r) we have

F ∗(�) = M∗(�) diag{(�− �0)m1 , . . . , (�− �0)mg , 1, . . . , 1}N∗(�).

where if g = n there are no ones appearing in the diagonal. And since M∗, N∗ ∈ O(�0) with

M∗(�0)) = M(�0)∗, N∗(�0)) = N(�0)∗ are invertible, this implies that G∗(�) = diag{(� −

�0)m1 , . . . , (�−�0)mg , 1, . . . , 1} is the local Smith form of F ∗ corresponding to the eigenvalue

�0. Therefore, g,m1 ≥ ⋅ ⋅ ⋅ ≥ mg,m are the geometric, partial, and algebraic multiplicities,

respectively, of the eigenvalue �0 of F ∗. This completes the proof.

Definition 15 Let U ⊆ ℂ2. Define U∗ := {(�, �) ∣ (�, �) ∈ U}. If L ∈ O(U,Mn(ℂ))

where U is an open connected set then the matrix function L∗ : U∗ → Mn(ℂ) defined by

L∗(�, �) = L(�, �)∗ is called the adjoint of the holomorphic matrix function L.

Lemma 26 If L ∈ O(U,Mn(ℂ)) where U ⊆ ℂ2 is an open connected set then U∗ is an open

connected set and L∗ ∈ O(U∗,Mn(ℂ)).

Proof. Let L ∈ O(U,Mn(ℂ)) where U ⊆ ℂ2 is an open connected set. Obviously U∗ is

an open connected set. The adjoint of L is L∗(�, �) := L(�, �). Now for any (�0, �0) ∈ U∗

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there exist (�0, z0) ∈ U such that (�0, �0) = (�0, z0). We define F1(�) := L(�, z0) = L∗(�, �0)

and F2(�) := L(�0, �) = L∗(�0, �). It follows from the one variable case that F1 ∈ O(�0) and

F2 ∈ O(�0). This implies the partial derivatives of L∗(�, �) exist at (�0, �0). But since this

is true for every (�0, �0) ∈ U∗ then L∗ ∈ O(U∗,ℂ). This completes the proof.

Lemma 27 Let L ∈ O(U,Mn(ℂ)). Then �(z) ∈ �(L(⋅, z)) if and only if �(�) ∈ �(L∗(⋅, �)),

where � = z. Moreover,

�(L∗) = �(L)∗. (3.61)

Proof. If �(z) ∈ �(L(⋅, z)) then 0 = detL(�(z), z) = detL(�(z), z)∗ = detL∗(�(z), z) =

detL∗(�(�), �) where � = z. Hence �(�) ∈ �(L∗(⋅, �)). Conversely, if �(�) ∈ �(L∗(⋅, �))

then letting � = z we have 0 = detL∗(�(�), �) = detL∗(�(z), z) = detL(�(z), z)∗ =

detL(�(z), z). Hence �(z) ∈ �(L(⋅, z)). The latter part of this statement follows from

the identity detL∗(�, �) = detL(�, �)∗ = detL(�, �) for any (�, �) ∈ U∗. This completes the

proof.

Definition 16 Suppose that �(z) is a convergent Puiseux series given by

�(z) =∞∑s=0

cs(z − z0)sq , ∣z − z0∣ < r (3.62)

where {cs}∞s=0 ⊆ ℂ. Then a convergent Puiseux series is given by

�∗(�) :=∞∑s=0

cs(� − z0)sq , ∣� − z0∣ < r. (3.63)

We call the Puiseux series �∗(�) the adjoint of the Puiseux series �(z).

Lemma 28 Let L ∈ O(U,Mn(ℂ)). Then �(z) is an eigenvalue Puiseux series of L(⋅, z) if

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and only if �∗(�) is an eigenvalue Puiseux series of L∗(⋅, �), where � = z. Moreover, if �ℎ(z),

ℎ = 0, . . . , q− 1 are all the branches of �(z) and �∗ℎ(�), ℎ = 0, . . . , q− 1 are all the branches

of �∗(�) then for � = z we have

q−1∪ℎ=0

{�∗ℎ(�)} =

q−1∪ℎ=0

{�ℎ(�)}, (3.64)

q−1∪ℎ=0

{�ℎ(z)} =

q−1∪ℎ=0

{�∗ℎ(z)}. (3.65)

Proof. Its obvious that

�(z) = �0 +∞∑s=1

cs(z − z0)sq , ∣z − z0∣ < r (3.66)

is a convergent Puiseux series if and only if its adjoint

�∗(�) := �0 +∞∑s=1

cs(� − z0)sq , ∣� − z0∣ < r (3.67)

is a convergent Puiseux series since (�∗)∗(z) = �(z). Denote the principal branch of the qth

root by q√z − z0 := q

√∣z − z0∣e

1qiarg(z−z0) for 0 ≤ arg(z− z0) ≤ 2� and let � := ei2�/q. Denote

another branch of the qth root by (z−z0)1/q := q√∣z − z0∣e

1qiarg(z−z0) for −� ≤ arg(z−z0) < �

and notice � = e−i2�/q is a primitive qth root of unity. A quick calculation shows

q√z − z0 = (z − z0)1/q, (3.68)

(z − z0)1/q = q√z − z0, (3.69)

for every z ∈ ℂ.

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Now all the branches of �(z) and �∗(�) are given by the convergent series

�ℎ(z) = �0 +∞∑s=1

cs(�ℎ q√z − z0)s, ℎ = 0, . . . , q − 1 ∣z − z0∣ < r,

�∗ℎ(�) = �0 +∞∑s=1

cs(�ℎ(� − z0)1/q)s, ℎ = 0, . . . , q − 1, ∣� − z0∣ < r.

Making the substituitions � = z and z = � as input to those branches and then using (3.68)

and (3.69), we prove the identities

�ℎ(z) = �∗ℎ(z), ℎ = 0, . . . , q − 1, ∣z − z0∣ < r, (3.70)

�∗ℎ(�) = �ℎ(�), ℎ = 0, . . . , q − 1, ∣� − z0∣ < r. (3.71)

This proves

q−1∪ℎ=0

{�∗ℎ(�)} =

q−1∪ℎ=0

{�ℎ(�)},q−1∪ℎ=0

{�ℎ(z)} =

q−1∪ℎ=0

{�∗ℎ(z)}

which is just the set of all values of the Puiseux series �∗(�) and �(z) at � and z, respectively,

where � = z. And hence, since for any convergent Puiseux series with say period q, the set

of all values at a point in its domain is independent of our choice of the branch of the qth

root function and primitive qth root of unity that we use, this proves the latter part of the

statement of this lemma. It now follows this and Lemma 27 that the values of the branches

of �(z) are eigenvalues of L(⋅, z) if and only if the branches of its adjoint �∗(�) are eigenvalues

of L∗(⋅, �), where � = z. And hence �(z) is an eigenvalue Puiseux series of L(⋅, z) if and only

if �∗(�) is an eigenvalue Puiseux series of L∗(⋅, �), where � = z. This completes the proof.

On Results from the Spectral Perturbation Theory. We are now ready to begin

giving some of the major results from [23, 34] that we will need in our thesis. The following

definition comes from [34].

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Definition 17 Let L ∈ O(U,Mn(ℂ)) and (�0, z0) ∈ U . Let (�(z), '(z)) be an eigenpair

Puiseux series of L(⋅, z) such that �(z) and '(z) are expanded about z0 with limit point �0

and �0, respectively. The vector �0 is called a generating eigenvector of L(⋅, z) (at the

point (�0, z0) and associated with �(z)).

Remark. It should be noted that in the case L ∈ O(U,Mn(ℂ)) with (�0, z0) ∈ U and �0 is

an eigenvalue of L(⋅, z0) of finite algebraic multiplicity, that by Theorem 23 there exists an

eigenpair Puiseux series (�(z), '(z)) of L(⋅, z) such that �(z) and '(z) are expanded about

z0 with limit point �0 and �0, respectively, for some �0. And hence a generating eigenvector

�0 of L(⋅, z) at the point (�0, z0) and associated with �(z) exists. And, in particular, we

have �0 ∕= 0 and �0 ∈ ker(L(�0, z0)). Moreover, by Lemma 28 and Lemma 22 it follows

there exists an eigenpair Puiseux series (�∗(�), (�)) of L∗(⋅, �) such that �∗(�) and (�)

are expanded about z0 with limit point �0 and 0, respectively, for some 0. Thus 0 is

a generating eigenvector of L∗(⋅, �) at the point (�0, z0) and associated with �∗(�), where

�∗(�) is the adjoint of the Puiseux series �(z) and L∗ is the adjoint of L. And, in particular,

0 ∕= 0 and 0 ∈ ker(L∗(�0, z0)) = kerL(�0, z0)∗.

The next proposition is equivalent to Lemma 7 from [34] and its proof comes from [23].

As we will shall soon see this proposition is actually one of the key results in the spectral

perturbation theory of holomorphic matrix functions.

Proposition 29 Let L ∈ O(U,Mn(ℂ)) and (�0, z0) ∈ U . Let �1(z), �2(z) be any two eigen-

value Puiseux series of L expanded about z0 with common limit point �0. Suppose that for

some branch �1,ℎ1(z) of �1(z) and some branch �2,ℎ2(z) of �2(z) there existed an r > 0 such

that �1,ℎ1(z) ∕= �2,ℎ2(z) for ∣z− z0∣ < r. Then for any generating eigenvector �0 of L(⋅, z) at

the point (�0, z0) and associated with �1(z) and for any generating eigenvector 0 of L∗(⋅, �)

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at the point (�0, z0) and associated with �∗2(�) we have

⟨L�(�0, z0)�0, 0⟩ℂ = 0. (3.72)

Proof. By assumption there exists eigenvalue Puiseux series

�1(z) =∞∑s=0

c1,s(z − z0)sq1 , ∣z − z0∣ < r

�2(z) =∞∑s=0

c2,s(z − z0)sq2 , ∣z − z0∣ < r

of L(⋅, z) such that for some fixed branch of the q1th and q2th root functions say f1 and f2

with its evaluation at (z−z0) given by f1(z−z0) and f2(z−z0), respectively, and some fixed

primitive q1th and q2th root of unity say �1 and �2, respectively, we have all their branches

given by the convergent series

�1,ℎ(z) = �0 +∞∑s=1

c1,s(�ℎ1f1(z − z0))s, ℎ = 0, . . . , q1 − 1, ∣z − z0∣ < r

�2,ℎ(z) = �0 +∞∑s=1

c2,s(�ℎ2f2(z − z0))s, ℎ = 0, . . . , q2 − 1, ∣z − z0∣ < r

and for some ℎ1 ∈ {0, . . . , q1 − 1} and ℎ2 ∈ {0, . . . , q2 − 1},

�1,ℎ1(z) ∕= �2,ℎ2(z), ∣z − z0∣ < r.

Now let �0 be a generating eigenvector of L(⋅, z) at the point (�0, z0) and associated with

�1(z) and 0 be a generating eigenvector of L∗(⋅, �) at the point (�0, z0) and associated

with �∗2(�). By definition of generating eigenvector, there exists an eigenpair Puiseux series

(�(z), '(z)) of L(⋅, z) such that �(z) and '(z) are expanded about z0 with limit point �0 and

�0, respectively, and there exists an eigenpair Puiseux series (�∗2(�), (�)) of L∗(⋅, �) such

85

that �(z) and '(z) are expanded about z0 with limit point �0 and 0, respectively.

Now because of the definition of eigenvector Puiseux series and the definition of adjoint of

a Puiseux series and because none of the properties just mentioned change by going to a

smaller domain for z, i.e., taking r smaller, we can assume without loss of generality that all

the branches of '(z), �∗2(�), (�) are given by the convergent series

'ℎ(z) = �0 +∞∑s=1

�1,s(�ℎ1f1(z − z0))s, ℎ = 0, . . . , q1 − 1, ∣z − z0∣ < r,

�∗2,ℎ(�) = �0 +∞∑s=1

c2,s(�ℎ2f2(� − z0))s, ℎ = 0, . . . , q2 − 1, ∣� − z0∣ < r,

ℎ(�) = �0 +∞∑s=1

�s(�ℎ2f2(� − z0))s, ℎ = 0, . . . , q2 − 1, ∣� − z0∣ < r,

respectively, and satisfy

L(�1,ℎ(z), z)'ℎ(z) = 0, ℎ = 0, . . . , q1 − 1, ∣z − z0∣ < r,

L∗(�∗2,ℎ(�), �) ℎ(�) = 0, ℎ = 0, . . . , q2 − 1, ∣� − z0∣ < r.

Now it follows from Lemma 28 that

q2−1∪ℎ=0

{�∗2,ℎ(�)} =

q2−1∪ℎ=0

{�2,ℎ(�)}, ∣� − z0∣ < r.

Thus for every � ∈ B(z0, r) there exists ℎ(�) ∈ {0, . . . , q2 − 1} such that

�∗2,ℎ(�)(�) = �2,ℎ2(�).

Hence if ∣z − z0∣ < r then for � = z we have

0 = L∗(�∗2,ℎ(�)(�), �) ℎ(�)(�) = L(�2,ℎ2(�), �)∗ ℎ(�)(�) = L(�2,ℎ2(z), z)∗ ℎ(z)(z)

86

and thus

⟨ ℎ(z)(z), L(�2,ℎ2(z), z)'ℎ1(z)⟩ℂ = ⟨L(�2,ℎ2(z), z)∗ ℎ(z)(z), 'ℎ1(z)⟩ℂ = 0.

Now it follows from Lemma 17, since L ∈ O(U,Mn(ℂ)) and (�0, z0) ∈ U , that L is an-

alytic at (�, z) = (�0, z0), i.e., the partial and mixed partial derivatives L(j1,j2)(�0, z0) :=

∂j1+j2L∂j1�∂j2z

∣∣(�,z)=(�0,z0)

exist for every j1, j2 ∈ ℕ ∪ {0} and there exists an r0 > 0 such that

L(�, z) =∞∑

j1,j2=0

1

j1!j2!L(j1,j2)(�0, z0)(�− �0)j1(z − z0)j2 , ∣�− �0∣ < r0, ∣z − z0∣ < r0

where the power series on the right converges absolutely in the Mn(ℂ) norm to L(�, z) for

(�, z) ∈ B(�0, r0)×B(z0, r0). We may assume without loss of generality that r0 ≤ r.

Thus from the facts above and the facts

xj − yj = (x− y)

j−1∑l=0

xlyj−1−l, x, y ∈ ℂ, j ∈ ℕ

limz→z0�1,ℎ1(z) = limz→z0

�1,ℎ2(z) = �0

limz→z0 ℎ(z)(z) = limz→z0

ℎ(z) = 0, ℎ = 0, . . . , q2 − 1

it follows that

(�1,ℎ1(z)− �0)j − (�2,ℎ2(z)− �0)j

�1,ℎ1(z)− �2,ℎ2(z)= o(1), as z → z0, if j ≥ 2

and so

L(�1,ℎ1(z), z)− L(�2,ℎ2(z), z)

�1,ℎ1(z)− �2,ℎ2(z)= L(1,0)(�0, z0) + o(1), as z → z0

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which implies

0 =⟨ ℎ(z)(z), L(�1,ℎ1(z), z)'ℎ1(z)⟩ℂ − ⟨ ℎ(z)(z), L(�2,ℎ2(z), z)'ℎ1(z)⟩ℂ

�1,ℎ1(z)− �2,ℎ2(z)

=

⟨ ℎ(z)(z),

L(�1,ℎ1(z), z)− L(�2,ℎ2(z), z)

�1,ℎ1(z)− �2,ℎ2(z)'ℎ1(z)

⟩ℂ

= ⟨ 0, L(1,0)(�0, z0)�0⟩ℂ + o(1), as z → z0.

Hence we must have

0 = ⟨ 0, L(1,0)(�0, z0)�0⟩ℂ = ⟨ 0, L�(�0, z0)�0⟩ℂ = ⟨L�(�0, z0)�0, 0⟩ℂ,

where by our notation L�(�0, z0) := L(1,0)(�0, z0). Therefore ⟨L�(�0, z0)�0, 0⟩ℂ = 0. This

completes the proof.

Lemma 30 Let �(z) be a convergent Puiseux series expanded about z0 with domain B(z0, r0)

and period q. Suppose �(z) is not a single-valued analytic function of z in B(z0, r0). Then

if we let �1(z), . . . , �q−1(z) be all the branches of �(z) with respect to a fix branch of the qth

root function and a fix primitive qth root of unity, there exists ℎ1, ℎ2 ∈ {0, . . . , q− 1} and an

r > 0 with r ≤ r0 such that

�ℎ1(z) ∕= �ℎ2(z), 0 < ∣z − z0∣ < r. (3.73)

Proof. Suppose �(z) is a convergent Puiseux series expanded about z0 with domain

B(z0, r0) and period q, where

�(z) =∞∑s=0

cs(z − z0)sq , ∣z − z0∣ < r0,

and suppose that �(z) is not a single-valued analytic function of z in B(z0, r0), i.e., there

exists s ∈ ℕ such that sq∕∈ ℕ and cs ∕= 0. We will denote by s0 the smallest such s with that

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property. Note that for the Puiseux series �(z) with the series representation give above we

must have the period q ≥ 2.

Now fix a branch of the qth root and denote its evaluation at z − z0 by (z − z0)1/q and fix

some primitive qth root of unity denoting it by �. Then the branches of �(z) are given by

the convergent series

�ℎ(z) =∞∑s=0

cs(�ℎ(z − z0)1/q)s, ∣z − z0∣ < r0, ℎ = 0, . . . , q − 1.

By the definition of s0 there exists an l ∈ ℕ ∪ {0} such that s0−1q

= l, cs0 ∕= 0, and

�ℎ(z) =l∑

j=0

cjq(z − z0)j + cs0(�ℎ(z − z0)1/q)s0 +

∞∑s=s0+1

cs(�ℎ(z − z0)1/q)s,

for ∣z− z0∣ < r0 and ℎ = 0, . . . , q− 1. It follows from this that for any ℎ1, ℎ2 ∈ {0, . . . , q− 1}

we have

∣�ℎ1(z)− �ℎ2(z)∣ = ∣cs0∣∣�ℎ1s0 − �ℎ2s0 ∣∣z − z0∣s0q +O(∣z − z0∣

s0+1q ), as z → z0.

The leading order term here can only be zero if ∣�ℎ1s0 − �ℎ2s0∣ = 0 which would imply

�(ℎ1−ℎ2)s0 = 1 and since � is a primitive qth root of unity this implies (ℎ1−ℎ2)s0q

∈ ℤ. Thus

letting ℎ1 := 1, ℎ2 := 0 ∈ {0, . . . , q − 1} (since q ≥ 2) then, because (ℎ1−ℎ2)s0q

= s0q∕∈ ℤ, we

have ∣�ℎ1s0 − �ℎ2s0∣ ∕= 0 and so

∣�ℎ1(z)− �ℎ2(z)∣∣cs0 ∣∣�ℎ1s0 − �ℎ2s0 ∣∣z − z0∣

s0q

= 1 + o(1), z ∕= z0, as z → z0.

Therefore, since limz→z0

�ℎ(z) = c0 for ℎ = 0, . . . , q− 1, it follows there exists r > 0 with r ≤ r0

such that

�ℎ1(z) ∕= �ℎ2(z), 0 < ∣z − z0∣ < r.

89

This completes the proof.

The following theorem is Theorem 9 from [34].

Theorem 31 Let L ∈ O(U,Mn(ℂ)) and suppose (�0, z0) ∈ U . Let �(z) be an eigenvalue

Puiseux series of L(⋅, z) expanded about z0 with limit point �0. Assume that for every gen-

erating eigenvector � of L(⋅, z) at the point (�0, z0) and associated with �(z) there exists a

generating eigenvector of L∗(⋅, �) at the point (�0, z0) and associated with �∗(�) such that

⟨L�(�0, z0)�, ⟩ℂ ∕= 0, (3.74)

where �∗(�) denotes the adjoint of �(z) and L∗ denotes the adjoint of L. Then �(z) is a

single-valued analytic function of z and there exists an eigenvector Puiseux series '(z) of

L(⋅, z) corresponding to �(z) which is also a single-valued analytic function of z.

Proof. Let L ∈ O(U,Mn(ℂ)) and suppose (�0, z0) ∈ U . Let �(z) be an eigenvalue

Puiseux series of L(⋅, z) expanded about z0, domain B(z0, r0), with limit point �0, and

period q. Suppose the hypotheses of this theorem are true for �(z). But suppose that �(z)

was not a single-valued analytic function of z ∈ B(z0, r0). Then by Lemma 30, if we let

�1(z), . . . , �q−1(z) be all the branches of �(z) with respect to a fix branch of the qth root

function and a fix primitive qth root of unity, there exists ℎ1, ℎ2 ∈ {0, . . . , q − 1} and an

r > 0 with r ≤ r0 such that

�ℎ1(z) ∕= �ℎ2(z), 0 < ∣z − z0∣ < r. (3.75)

Now by Lemma 22 there exists an eigenvector Puiseux series '(z) of L(⋅, z) corresponding

to the eigenvalue Puiseux series �(z). Let �0 denote the limit point of '(z) when expanded

about z0. By definition �0 is a generating eigenvector of L(⋅, z) at the point (�0, z0) and

90

associated with �(z) and so by our hypotheses their exists a eigenvalue Puiseux series (�),

expanded about z0 with limit point 0, of L∗(⋅, �) corresponding to the eigenvalue Puiseux

series �∗(�) such that

⟨L�(�0, z0)�0, 0⟩ℂ ∕= 0. (3.76)

But according to Proposition 29 with �1(z) := �2(z) = �(z) we must have

⟨L�(�0, z0)�0, 0⟩ℂ = 0. (3.77)

This is a contradiction. Therefore, �(z) is single-valued analytic function of z ∈ B(z0, r0)

and so by Lemma 22 there exists an eigenvector Puiseux series '(z) of L(⋅, z) corresponding

to �(z) which is also a single-valued analytic function of z. This completes the proof.

The following proposition comes from [23, Lemma 4.1].

Proposition 32 Let L ∈ O(U,Mn(ℂ)) and suppose (�0, z0) ∈ �(L). Assume (�(z), '(z)) is

an eigenpair Puiseux series of L(⋅, z) such that both �(z) and '(z) are analytic functions at

z0 with �(z0) = �0. Denote the order of the zero of the function �(z)− �0 at z0 by q. Then

'(z) is a generating function of order q for L(�0, ⋅) at the eigenvalue z0.

Proof. By hypotheses there exists an r > 0 such that B(�0, r) × B(z0, r) ⊆ U , �(⋅) ∈

O(B(z0, r),ℂ), and '(⋅) ∈ O(B(z0, r),ℂn) such that '(z0) ∕= 0 and L(�(z), z)'(z) = 0 for

every z ∈ B(z0, r). Thus it follows from this by using the power series expansion of L at

(�0, z0) from Lemma 17, that

'(z0) ∕= 0, L(�0, z)'(z) = L(�0, z)'(z)− L(�(z), z)'(z)

= (L(�0, z)− L(�(z), z))'(z)

= −(�(z)− �0)L!(�0, z0)'(z) + o(�(z)− �0)

91

= −�(q)(z0)(z − z0)qL!(�0, z0)'(z0) + o((z − z0)q)

= O((z − z0)q), as z → z0.

By definition of generating function this implies '(z) is a generating function of order q for

L(�0, ⋅) at the eigenvalue z0.

92

Chapter 4

Canonical Equations: A Model for

Studying Slow Light

4.1 Introduction

In this chapter we consider a general nondissipative but dispersive model for wave propa-

gation using an important class of periodic differential and differential-algebraic equations

(DAEs) called canonical equations [31]. This model is general enough to include electromag-

netic waves governed by the time-harmonic Maxwell’s equations for lossless one-dimensional

photonic crystals whose constituent layers can be any combination of isotropic, anisotropic,

or bianisotropic materials with or without material dispersion (i.e., frequency-dependent re-

sponse of materials). This makes our work particularly significant in the study of slow light

since metamaterials are widening the range of potential photonic crystals that can be fab-

ricated and so a model like ours that has the ability to analysis slow light phenomena for a

broad range of photonic crystals is in need.

93

4.1.1 Model Formulation

In this chapter1, we will denoted a wave by y and its evaluation at a position x by y(x),

where x ∈ ℝ, y(x) ∈ ℂN , and y ∈ (L2loc(ℝ))N . The frequency domain Ω is an open connected

set in ℂ and the real frequency domain Ωℝ := Ω ∩ ℝ is nonempty.

Wave propagation will be governed by canonical systems of differential equations with peri-

odic coefficients that depend holomorphically on the frequency parameter !, i.e., differential

equations of the form

G y′(x) = V (x, !)y(x), (4.1)

where the leading matrix coefficient G ∈MN(ℂ) and the matrix-valued function V : ℝ×Ω→

MN(ℂ), the Hamiltonian, have the properties:

(i) G ∗ = −G

(ii) V (x, !)∗ = V (x, !), for each ! ∈ Ωℝ and almost every x ∈ ℝ

(iii) V (x+ d, !) = V (x, !), for each ! ∈ Ω and almost every x ∈ ℝ

(iv) V ∈ O(Ω,MN(Lp(T))) as a function of frequency where

p =

⎧⎨⎩ 1 if det(G ) ∕= 0,

2 if det(G ) = 0.

In the case det(G ) = 0, the domain of equation (4.1) and definition of its leading term are

D := {y ∈ (L2loc(ℝ))N : P⊥y ∈ (W 1,1

loc (ℝ))N}, G y′ := G (P⊥y)′, (4.2)

1See section 4.5 for notation and convention.

94

where P , P⊥ ∈ MN(ℂ) denote the projections onto the kernel and range of G , respectively.

In the case det(G ) ∕= 0, we take the standard domain D := (W 1,1loc (ℝ))N and definition of y′

as the weak derivative of the entries of y.

The equations of the form (4.1) with det(G ) = 0 are differential-algebraic equations (DAEs)

and are not just ODEs. Their solutions must satisfy an algebraic equation as well, namely,

PV (x, !)y(x) = 0 for a.e. x ∈ ℝ. Thus, in order to guarantee the existence of a nontrivial

solution for each frequency and a solution space which depends holomorphically on frequency,

we impose an additional hypothesis for equations of the form (4.1) whenever det(G ) = 0,

namely, (G + PV (x, !)P )−1 exists for each ! ∈ Ω and a.e. x ∈ ℝ and as a function of

frequency

(G + PV P )−1 ∈ O(Ω,MN(L∞(0, d))). (4.3)

We will refer to this as the index-1 hypothesis.

Differential-algebraic equations of the form given by (4.1), for real frequencies, are known as

canonical equations [31] and, in the case det(G ) ∕= 0, they are known as canonical differential

equations, linear canonical systems, linear Hamiltonian systems, or Hamiltonian equations.

We distinguish between the two classes of canonical equations of the form (4.1) by referring

to those with det(G ) = 0 as canonical DAEs and those with det(G ) ∕= 0 as canonical ODEs.

We introduce the following definitions into our model:

Definition 18 We say y is a solution of equation (4.1) provided y ∈ D and equation (4.1)

is satisfied for some ! ∈ Ω and for a.e. x in ℝ. In which case we call ! its frequency.

Definition 19 We say y is a Floquet solution provided for some ! ∈ Ω, it is a solution

95

of equation (4.1) with ! as its frequency and has the Floquet representation

y(x) = eikxm∑j=0

xjuj(x), (4.4)

where m ∈ ℕ∪{0} and uj ∈ D∩ (Lp(T))N , for j = 1, . . . ,m. We call k, (k, !), and � := eikd

its wavenumber, wavenumber-frequency pair, and Floquet multiplier, respectively.

If um ∕= 0 then we call m its order. We call y a Bloch solution if it is a Floquet solution

with m = 0. If, in addition, y is a Bloch solution with real frequency then we call it a

propagating wave if it has a real wavenumber and an evanescent wave otherwise.

Definition 20 We define the Bloch variety to be the set

ℬ := {(k, !) ∈ ℂ× Ω ∣ (k, !) is the wavenumber-frequency pair of some (4.5)

nontrivial Bloch solution of equation (4.1)}

and the real Bloch variety to be the set

ℬℝ := ℬ ∩ ℝ2. (4.6)

Definition 21 We define the dispersion relation denoted by ! = !(k) as the multi-

valued function whose graph is the Bloch variety. We say a point (k0, !0) ∈ ℬℝ is amicable

if for every � > 0 there exists an � > 0 such that ℬ ∩ B(k0, �) × B(!0, �) is the union

of the graphs of a finite number of real analytic functions, say !1(k), . . . , !g(k), with the

property !1(k0) = ⋅ ⋅ ⋅ = !g(k0) = !0. We call these functions the band functions and

their derivatives d!1

dk, . . . , d!g

dkthe group velocities. The intersection of the graph of a band

function with ℝ2 is called a band.

Definition 22 For any solution y of equation (4.1) with real frequency !0 we define its

96

energy flux and energy density to be the functions of position given by

⟨iG y, y⟩ℂ (4.7)

⟨V!(⋅, !0)y, y⟩ℂ (4.8)

respectively, where V! denotes the derivative of the Hamiltonian V with respect to frequency in

the MN(Lp(T)) norm. We define its average energy flux and average energy density

to be the averages

1

d

∫ d

0

⟨iG y(x), y(x)⟩ℂdx (4.9)

1

d

∫ d

0

⟨V!(x, !0)y(x), y(x)⟩ℂdx (4.10)

respectively. We also define its energy velocity as the ratio of its average energy flux to

its average energy density provided the latter is nonzero, i.e.,

1d

∫ d0⟨iG y(x), y(x)⟩ℂdx

1d

∫ d0⟨V!(x, !0)y(x), y(x)⟩ℂdx

. (4.11)

Remark: The integrals (4.9) and (4.10) are well-defined because Corollary 40, Corollary 43,

and Theorem 50 show the functions (4.7) and (4.8) are in L1loc(ℝ).

Definition 23 We say (k0, !0) ∈ ℬℝ is a point of definite type for the canonical equations

in (4.1) and say these equations are of definite type at this point provided the average energy

density of any nontrivial Bloch solution y with (k0, !0) as its wavenumber-frequency pair is

nonzero, i.e.,

1

d

∫ d

0

⟨V!(x, !0)y(x), y(x)⟩ℂdx ∕= 0. (4.12)

97

The canonical equations in (4.1) along with the definitions just given constitute the model

of wave propagation that we will analyze in this thesis. Moreover, based on our studies

on electromagnetic wave propagation in one-dimensional photonic crystals, it is found by

physical considerations that a point of definite type is the right assumption in which to

begin a perturbation analysis from stationary points of the dispersion relation.

We are particularly interested in the perturbation analysis of canonical equations in the slow

wave regime in which points in the Bloch variety (k, !) belong to a small neighborhood of

a point (k0, !0) in the real Bloch variety where (k0, !0) is both a point of definite type for

the canonical equations (such a point is amicable (see Theorem 48 and Theorem 50)) and a

stationary point of the dispersion relation, i.e., at (k0, !0) one of the bands has a stationary

point. For this model and with this regime in mind, we give the following definition for a

slow wave:

Definition 24 Let (k0, !0) be any point which is amicable and 0 < r ≪ 1. Then any

propagating Bloch wave y with wavenumber-frequency pair (k, !) belonging to a band with

a stationary point at (k0, !0) and satisfying 0 < ∣∣(k, !) − (k0, !0)∣∣ℂ < r is called a slow

wave.

The main purpose of this chapter is to begin developing a mathematical framework for this

model, including the Floquet, spectral, and perturbation theory, and to use this framework

to analyze in detail the slow wave regime.

4.2 Canonical ODEs

In this section we consider the canonical ODEs

J ′(x) = A(x, !) (x), ∈ (W 1,1loc (ℝ))n (4.13)

98

where, as discussed in section 4.1.1, the leading matrix coefficient J ∈ Mn(ℂ) and the

Hamiltonian A : ℝ× Ω→Mn(ℂ) have the properties:

(i) det(J ) ∕= 0, J ∗ = −J ,

(ii) A(x, !)∗ = A(x, !), for each ! ∈ Ωℝ and a.e. x ∈ ℝ,

(iii) A(x+ d, !) = A(x, !), for each ! ∈ Ω and a.e. x ∈ ℝ,

(iv) A ∈ O(Ω,MN(L1(T))) as a function of frequency.

Denote by A! the derivative of the Hamiltonian A with respect to frequency in theMn(L1(T))

norm.

4.2.1 Preliminaries

We begin with some preliminary results on the solutions of the canonical ODEs in (4.13)

and the Floquet theory relating to the matricant and monodromy matrix of these canoni-

cal ODEs. The dependency of the matricant and monodromy matrix on frequency is also

discussed. We will hold off on proving the results of this subsection until section 4.4. The

statements in this subsection are known, but we state them here in order to setup the main

results of this chapter. We also will give proofs of these statements for completeness in

section 4.4.

Proposition 33 For each ! ∈ Ω, let Ψ(⋅, !) denote the fundamental matrix solution of

the canonical ODEs in (4.13) satisfying Ψ(0, !) = In, i.e., the unique function Ψ(⋅, !) ∈

Mn(W 1,1loc (ℝ)) satisfying a.e. the matrix differential equation with initial condition

J Ψ′(x) = A(x, !)Ψ(x), Ψ(0) = In. (4.14)

99

Then the matrix-valued function Ψ : ℝ× Ω→Mn(ℂ) has the following properties:

(i) is a solution of the canonical ODEs in (4.13) at the frequency ! ∈ Ω if and only if

= Ψ(⋅, !) for some ∈ ℂn. Moreover, is unique.

(ii) For every (x, !) ∈ ℝ× Ω,

Ψ(x, !) = In +

∫ x

0

J −1A(t, !)Ψ(t, !)dt. (4.15)

(iii) For every x ∈ ℝ, Ψ(x, ⋅) ∈ O(Ω,Mn(ℂ)). We will denote the partial derivative of the

function Ψ with respect to frequency in the Mn(ℂ) norm by Ψ!.

(iv) For every (x, !) ∈ ℝ× Ω, Ψ!(⋅, !) ∈Mn(W 1,1loc (ℝ)) and

Ψ!(x, !) =

∫ x

0

J −1A!(t, !)Ψ(t, !) + J −1A(t, !)Ψ!(t, !)dt (4.16)

(v) For every (x, !) ∈ ℝ×Ω, Ψ−1(x, !) := Ψ(x, !)−1 exists and Ψ−1(⋅, !) ∈Mn(W 1,1loc (ℝ)).

(vi) For every (x, !) ∈ ℝ× Ω,

Ψ(x+ d, !) = Ψ(x, !)Ψ(d, !). (4.17)

Now for each fixed ! ∈ Ω, Ψ(⋅, !) and Ψ(d, !) are the matricant and monodromy matrix,

respectively, of the canonical ODEs in (4.13). Thus the previous proposition implies the

following:

Theorem 34 (Floquet-Lyapunov) Let ! ∈ Ω. Then there exists a matrix K(!) ∈Mn(ℂ)

and a function F (⋅, !) ∈Mn(W 1,1loc (ℝ)) such that for every x ∈ ℝ,

Ψ(x, !) = F (x, !)eixK(!),

100

F (x + d, !) = F (x, !), F (0, !) = In, F−1(x, !) := F (x, !)−1 exists, and F−1(⋅, !) ∈

Mn(W 1,1loc (ℝ)).

Theorem 35 If is a Floquet solution of the canonical ODEs in (4.13) with wavenumber-

frequency pair (k, !), Floquet multiplier � = eikd, and order m then = Ψ(⋅, !) where

is a generalized eigenvector of Ψ(d, !) of order m + 1 corresponding to the eigenvalue �.

Conversely, if is a generalized eigenvector of order m+1 of the monodromy matrix Ψ(d, !)

corresponding to the eigenvalue � then for any k ∈ ℂ such that � = eikd, = Ψ(⋅, !) is

a Floquet solution of the canonical ODEs in (4.13) with wavenumber-frequency pair (k, !),

Floquet multiplier � = eikd, and order m.

Corollary 36 Let ! ∈ Ω. Let { j}nj=1 be a Jordan basis for the matrix Ψ(d, !). For

j = 1, . . . , n, let lj, �j, and kj be such that j is a generalized eigenvalue of Ψ(d, !) of

order lj corresponding to the eigenvalue �j = eikjd. Define { j}nj=1 by j := Ψ(⋅, !) j for

j = 1, . . . , n. Then the following statements are true:

(i) The set of solutions of the canonical ODEs in (4.13) at the frequency ! is a vector

space over ℂ and { j}nj=1 is a basis for this space.

(ii) For j = 1, . . . , n, j is a Floquet solution of the canonical ODEs in (4.13) with

wavenumber-frequency pair (kj, !), Floquet multiplier �j = eikjd, and order lj − 1.

Corollary 37 Let ℬ denote the Bloch variety of the canonical ODEs in (4.13). Then, for

any l ∈ ℤ,

ℬ = (2�l/d, 0) + ℬ. (4.18)

Corollary 38 Define the function D : ℂ× Ω→ ℂ by

D(k, !) := det(eikdIn −Ψ(d, !)

). (4.19)

101

Then D is a nonconstant holomorphic function and its zero set is the Bloch variety of the

canonical ODEs in (4.13), i.e.,

ℬ = {(k, !) ∈ ℂ× Ω ∣ D(k, !) = 0}. (4.20)

4.2.2 Energy Flux, Energy Density, and their Averages

The following results describe how the energy flux, energy density, their averages, and points

of definite type of the canonical ODEs in (4.13) are related to the leading matrix coefficient,

the matricant, and the monodromy matrix.

We start by considering the energy flux and its average. The following proposition is the key

result, as we will see in this chapter, in the study of energy flux for canonical equations.

Proposition 39 For each ! ∈ Ωℝ we have Ψ(⋅, !)∗iJ Ψ(⋅, !) ∈Mn(W 1,1loc (ℝ)) and

Ψ(⋅, !)∗iJ Ψ(⋅, !) = iJ . (4.21)

This proposition is important because it tells us the energy flux of any solution of canonical

ODEs with real frequency is a conserved quantity. A more precise statement of this is found

in the next corollary.

Corollary 40 For each ! ∈ Ωℝ and every 1, 2 ∈ ℂn, if 1 = Ψ(⋅, !) 1 and 2 = Ψ(⋅, !) 2

then ⟨iJ 1, 2⟩ℂ ∈ W1,1loc (ℝ) and

⟨iJ 1, 2⟩ℂ = ⟨iJ 1, 2⟩ℂ, (4.22)

1

d

∫ d

0

⟨iJ 1(x), 2(x)⟩ℂdx = ⟨iJ 1, 2⟩ℂ. (4.23)

102

We will now consider the energy density and its average.

Lemma 41 For each ! ∈ Ωℝ we have A!(⋅, !) ∈Mn(L1(T)) and

A!(⋅, !)∗ = A!(⋅, !). (4.24)

Theorem 42 For each ! ∈ Ωℝ we have

Ψ(⋅, !)∗J Ψ!(⋅, !) ∈Mn(W 1,1loc (ℝ)), (4.25)

Ψ(⋅, !)∗A!(⋅, !)Ψ(⋅, !) ∈Mn(L1loc(ℝ)), (4.26)

Ψ(⋅, !)∗A!(⋅, !)Ψ(⋅, !) = (Ψ(⋅, !)∗J Ψ!(⋅, !))′, (4.27)

1

d

∫ d

0

Ψ(t, !)∗A!(t, !)Ψ(t, !)dt =1

dΨ(d, !)∗J Ψ!(d, !). (4.28)

Corollary 43 For each ! ∈ Ωℝ and every 1, 2 ∈ ℂn, if 1 = Ψ(⋅, !) 1 and 2 = Ψ(⋅, !) 2

then ⟨Ψ(⋅, !)∗J Ψ!(⋅, !) 1, 2⟩ℂ ∈ W1,1loc (ℝ), ⟨A!(⋅, !) 1, 2⟩ℂ ∈ L1

loc(ℝ) and

⟨A!(⋅, !) 1, 2⟩ℂ = ⟨Ψ(⋅, !)∗J Ψ!(⋅, !) 1, 2⟩ℂ′ , (4.29)

1

d

∫ d

0

⟨A!(x, !) 1(x), 2(x)⟩ℂdx =1

d⟨Ψ(d, !)∗J Ψ!(d, !) 1, 2⟩ℂ. (4.30)

4.2.3 On Points of Definite Type for Canonical ODEs

Let (k, !) ∈ ℬℝ and on the eigenspace of the monodromy matrix Ψ(d, !) corresponding to

the eigenvalue eikd we define the sesquilinear form q(k,!)

by

q(k,!)

( 1, 2) := 1d

∫ d0⟨A!(x, !) 1(x), 2(x)⟩ℂdx (= 1

d⟨Ψ(d, !)∗J Ψ!(d, !) 1, 2⟩ℂ),

1, 2 ∈ ker(eikdIn −Ψ(d, !)), 1 = Ψ(⋅, !) 1, 2 = Ψ(⋅, !) 2.

(4.31)

It follows from Corollary 43 that q(k,!)

is well-defined.

103

Lemma 44 For each (k, !) ∈ ℬℝ, q(k,!)

is a Hermitian form, that is, q(k,!)

satisfies

q(k,!)

( 1, 2) = q(k,!)

( 2, 1) (4.32)

for every 1, 2 ∈ ker(eikdIn − Ψ(d, !)). In particular, q(k,!)

( , ) ∈ ℝ for every ∈

ker(eikdIn −Ψ(d, !)).

Denote the sign function by sgn(x) := x∣x∣ ∈ {−1, 1}, for x ∈ ℝ/{0}.

Proposition 45 The canonical ODEs in (4.13) are of definite type at a point (k0, !0) ∈ ℬℝ

if and only if q(k0,!0)

is a definite sesquilinear form which is bounded, i.e., the following

properties hold:

(i) For every nonzero ∈ ker(eikdIn −Ψ(d, !)) we have

q(k0,!0)

( , ) ∈ ℝ/{0} (4.33)

and its sign sgn(q(k0,!0)

( , )) =: sgn(q(k0,!0)

) is independent of the choice of .

(ii) The sesquilinear form

⟨ 1, 2⟩(k0,!0) := sgn(q(k0,!0)

)q(k0,!0)

( 1, 2),

1, 2 ∈ ker (eik0dIn −Ψ(d, !0))

(4.34)

is inner product on ker(eik0dIn −Ψ(d, !0)).

(iii) There exists constants C1, C2 > 0 such that

C1∣∣ ∣∣2ℂ ≤ ∣q(k0,!0)( , )∣ ≤ C2∣∣ ∣∣2ℂ, for all ∈ ker(eik0dIn −Ψ(d, !0)). (4.35)

104

Lemma 46 The family of sesquilinear forms {q(k,!)}(k,!)∈ℬℝ depends continuously on the

indexing parameter (k, !) in the following sense:

If {(kj, !j)}j∈ℕ ⊆ ℬℝ, { j}j∈ℕ ⊆ ℂn such that j ∈ ker(eikjdIn − Ψ(d, !j)) with ∣∣ j∣∣ℂ = 1

for all j ∈ ℕ and

(kj, !j)∣∣⋅∣∣ℂ−−→ (k0, !0) and j

∣∣⋅∣∣ℂ−−→ 0, as j →∞

then (k0, !0) ∈ ℬℝ, 0 ∈ ker(eik0dIn −Ψ(d, !0)) with ∣∣ 0∣∣ℂ = 1 and

q(kj,!j)

( j, j)∣⋅∣−→ q

(k0,!0)( 0, 0) as j →∞. (4.36)

Theorem 47 If (k0, !0) ∈ ℬℝ is a point of definite type for the canonical ODEs in (4.13)

then there exists an r > 0 such that every (k, !) ∈ B((k0, !0), r) ∩ ℬℝ is a point of definite

type and

sgn(q(k,!)

) = sgn(q(k0,!0)

). (4.37)

4.2.4 Perturbation Theory for Canonical ODEs

This section contains the main results of this chapter on the perturbation theory for canonical

ODEs.

Theorem 48 Suppose the canonical ODEs in (4.13) are of definite type at (k0, !0) ∈ ℬℝ.

Let g be the number of Jordan blocks (geometric multiplicity) in the Jordan form of the

monodromy matrix Ψ(d, !0) corresponding to the eigenvalue �0 = eik0d and m1 ≥ ⋅ ⋅ ⋅ ≥

mg ≥ 1 the dimensions of each of those Jordan blocks (partial multiplicities). Define m :=

m1 + ⋅ ⋅ ⋅ + mg (algebraic multiplicity). Let � > 0 be given. Then there exists an � > 0 such

105

that

(i) The order of the zero of D(k0, !) at ! = !0 is g and the order of the zero of D(k, !0)

at k = k0 is m.

(ii) The set ℬ∩B(k0, �)×B(!0, �) is the union of the graphs of g nonconstant real analytic

functions !1(k), . . . , !g(k) given by the convergent power series

!j(k) = !0 + �j,mj(k − k0)mj +∞∑

l=mj+1

�j,l(k − k0)l, ∣k − k0∣ < � (4.38)

where

�j,mj ∕= 0, (4.39)

for j = 1, . . . , g. Moreover, there exists analytic functions '1(k), . . . , 'g(k) belonging

to O(B(k0, �),ℂn) such that

'j(k) ∕= 0, Ψ(d, !j(k))'j(k) = eikd'j(k), ∣k − k0∣ < �, j = 1, . . . , g, (4.40)

the vectors '1(k), . . . , 'g(k) are linearly independent for ∣k−k0∣ < �, and, in particular,

the vectors '1(k0), . . . , 'g(k0) form a basis for ker(eik0dIn −Ψ(d, !0)).

Corollary 49 The conditions

D(k0, !0) = 0,∂D

∂!(k0, !0) ∕= 0, (k0, !0) ∈ ℝ× Ωℝ (4.41)

are satisfied if and only if (k0, !0) ∈ ℬℝ is a point of definite type for the canonical ODEs in

(4.13) and the Jordan normal form of the monodromy matrix Ψ(d, !0) corresponding to the

eigenvalue �0 = eik0d consists of a single Jordan block, i.e., dim ker(�0In −Ψ(d, !0)) = 1.

106

Remark. In this case, by denoting �0 := eik0d, the conditions of the previous corollary are

det(�0In − Ψ(d, !)) = D(k0, !) = 0 and ∂∂!

det(�In − Ψ(d, !))∣(!0,�0) = ∂D∂!

(k0, !0) ∕= 0,

but this is just the generic condition (2.1) in chapter 2 (see also [61, See (1.1)]). This

condition was throughly investigated in chapter 2 (see also [61, See (1.1)]) for general matrix

perturbations. And, in particular, in Theorem 2 of chapter 2 (see also [61, Theorem 3.1])

explicit recursive formulas are given for the unique eigenvalue Puiseux series �(!) which

satisfies �(!0) = �0 and for its associated eigenvector Puiseux series of the monodromy

matrix Ψ(d, !) near the frequency ! = !0. Moreover, their Puiseux series coefficients up to

the second order are conveniently listed in Corollary 4 of chapter 2 (see also [61, Corollary

3.3]).

4.3 Canonical DAEs

In this section we consider the canonical DAEs

G ′(x) = V (x, !)y(x), y ∈ D (4.42)

where, as discussed in §1, the leading matrix coefficient G ∈ Mn(ℂ) and the Hamiltonian

V : ℝ× Ω→MN(ℂ) have the properties:

(i) det(G ) = 0, G ∗ = −G ,

(ii) V (x, !)∗ = V (x, !), for each ! ∈ Ωℝ and a.e. x ∈ ℝ,

(iii) V (x+ d, !) = V (x, !), for each ! ∈ Ω and a.e. x ∈ ℝ,

(iv) V ∈ O(Ω,MN(L2(T))) as a function of frequency.

107

Denote by V! the derivative of the Hamiltonian V with respect to frequency in the Mn(L2(T))

norm.

The domain D of the canonical DAEs in (4.42) and definition of its leading term are

D := {y ∈ (L2loc(ℝ))N : P⊥y ∈ (W 1,1

loc (ℝ))N}, G y′ := G (P⊥y)′, (4.43)

where P , P⊥ ∈MN(ℂ) denote the projections onto the kernel and range of G , respectively.

We shall assume that the canonical DAEs in (4.42) satisfy the index-1 hypothesis, namely,

(G + PV P )−1 ∈ O(Ω,MN(L∞(0, d))). (4.44)

The goal of this section is to describe the theory of canonical DAEs including the Floquet

and spectral perturbation theory in terms of the theory developed in the previous section

for canonical ODEs. We will only present the statement of our main result, the proof and

consequences will appear in [60].

4.3.1 The Correspondence between Canonical DAEs and Canon-

ical ODEs

The following represents our main result in the study of canonical DAEs:

Theorem 50 Let n := dim ran(G ). Then there exists an n× n system of canonical ODEs

J ′(x) = A(x, !) (x), ∈ (W 1,1loc (ℝ))n (4.45)

with J ∈Mn(ℂ), A ∈ O(Ω,Mn(L1(T))), and a function Q ∈ O(Ω,MN×n(L2(T))) such that

the multiplication map Q(⋅, !) : (W 1,1loc (ℝ))n → D is injective and the following properties

108

hold:

(i) The solutions of the canonical DAEs in (4.42) with frequency ! are given by y =

Q(⋅, !) where is a solution of the canonical ODEs in (4.45) with frequency !.

(ii) If y = Q(⋅, !) then y is Floquet solution of the canonical DAEs with wavenumber-

frequency pair (k, !), Floquet multiplier � = eikd, and order m if and only if is a

Floquet solution of the canonical ODEs with wavenumber-frequency pair (k, !), Floquet

multiplier � = eikd, and order m.

(iii) If y = Q(⋅, !0) is a solution of the canonical DAEs with frequency !0 ∈ Ωℝ then its

energy flux and energy density are in L1loc(ℝ) and

⟨iG y, y⟩ℂ = ⟨iJ , ⟩ℂ, ⟨V!(⋅, !0)y, y⟩ℂ = ⟨A!(⋅, !0) , ⟩ℂ. (4.46)

To summarize this theorem, the study of canonical DAEs (with index-1 hypothesis), in-

cluding the Floquet, spectral, and perturbation theory, is reduced to the study of canonical

ODEs and the theory already developed in this chapter. In [60] we will elaborate on this

correspondence in more detail and discuss the spectral perturbation theory as well.

4.4 Proofs

Proofs for Section 4.2.1

We begin by introducing a differential operator associated to the canonical ODEs in (4.13).

We consider the frequency dependent operator

T (!) := ′ −J −1A(⋅, !) , ∈ (W 1,1loc (ℝ))n, ! ∈ Ω. (4.47)

109

Lemma 51 For each ! ∈ Ω, T (!) is a linear operator with T (!) : (W 1,1loc (ℝ))n → (L1

loc(ℝ))n.

Proof. We begin by proving the operator T (!) : (W 1,1loc (ℝ))n → (L1

loc(ℝ))n given by

(4.47) is well-defined. Let ∈ (W 1,1loc (ℝ))n. Then by definition of the space (W 1,1

loc (ℝ))n we

have ′ ∈ (L1loc(ℝ))n. We now show J −1A(⋅, !) ∈ (L1

loc(ℝ))n. To do this let (a, b) ⊆

ℝ be any bounded interval. It follows from Lemma 88 that A(⋅, !)∣(a,b) ∈ Mn(L1(a, b))

and from Lemma 69 that ∣(a,b) ∈ (W 1,1(a, b))n. It then follows from Lemma 83 that

A(⋅, !)∣(a,b) ∣(a,b) ∈ (L1(a, b))n and hence J −1A(⋅, !)∣(a,b) ∣(a,b) ∈ (L1(a, b))n. From this

it follows that (J −1A(⋅, !) )∣(a,b) = J −1A(⋅, !)∣(a,b) ∣(a,b) ∈ (L1(a, b))n for every bounded

interval (a, b) ⊆ ℝ, this implies J −1A(⋅, !) ∈ (L1loc(ℝ))n. And hence ′ −J −1A(⋅, !) ∈

(L1loc(ℝ))n. This proves that the operator T (!) : (W 1,1

loc (ℝ))n → (L1loc(ℝ))n given by (4.47) is

well-defined. Therefore, since its obvious the operator T (!) is linear, the proof is complete.

Lemma 52 is a solution of the canonical ODEs in (4.13) with frequency ! ∈ Ω if and

only if ∈ ker(T (!)).

Proof. Let is a solution of the canonical ODEs in (4.13) with frequency ! ∈ Ω. This

means by Definition 18 that ∈ (W 1,1loc (ℝ))n and for a.e. x ∈ ℝ satisfies the equation

J ′(x) = A(x, !) (x).

Hence for a.e. x ∈ ℝ satisfies the equation

′(x) = J −1A(x, !) (x).

110

But this implies, since as we showed in the proof of Lemma 4.47 that J −1A(⋅, !) ∈

(L1loc(ℝ))n, that T (!) = ′ −J −1A(⋅, !) = 0 in (L1

loc(ℝ))n. This proves ∈ ker(T (!)).

Conversely, if ∈ ker(T (!)) then ∈ (W 1,1loc (ℝ))n and 0 = T (!) = ′ −J −1A(⋅, !) in

(L1loc(ℝ))n. This implies that for a.e. x ∈ ℝ, satisfies the equation

′(x) = J −1A(x, !) (x),

or equivalently, the equation

J ′(x) = A(x, !) (x).

Therefore by Definition 18 this means is a solution of the canonical ODEs in (4.13) with

frequency ! ∈ Ω. This completes the proof.

Let (a, b) ⊆ ℝ be a bounded interval. Then we can consider (W 1,1(a, b))n and (L1(a, b))n as

a subspace of (W 1,1loc (ℝ))n and (L1

loc(ℝ))n, respectively, by identifying them with the image

of those latter spaces under the surjective map f 7→ f ∣(a,b). Then we can restrict the domain

of T (!) to the subspace (W 1,1(a, b))n and it becomes a differential operator on (W 1,1(a, b))n

with range in (L1(a, b))n. It is with this perspective in mind that we introduce and consider

the following frequency dependent operator

T ∣(a,b)(!) := ′ −J −1A(⋅, !)∣(a,b) , ∈ (W 1,1(a, b))n, ! ∈ Ω. (4.48)

Proposition 53 For each ! ∈ Ω we have

T ∣(a,b)(!) : (W 1,1(a, b))n → (L1(a, b))n (4.49)

111

T ∣(a,b)(!) ∈ L ((W 1,1(a, b))n, (L1(a, b))n) (4.50)

T ∣(a,b) ∈ O(Ω,L ((W 1,1(a, b))n, (L1(a, b))n)). (4.51)

Moreover,

(T (!) )∣(a,b) = T ∣(a,b)(!) ∣(a,b), (4.52)

for every ! ∈ Ω, every ∈ (W 1,1loc (ℝ))n, and every bounded interval (a, b) ⊆ ℝ.

Proof. To be begin we note that by hypothesis A ∈ O(Ω,Mn(L1(T))). Let (a, b) ⊆ ℝ)

be any bounded interval. It follows from Lemma 88 that for each ! ∈ Ω, A∣(a,b)(⋅, !) :=

A(⋅, !)∣(a,b) ∈Mn(L1(a, b)). It then follows from Lemma 93 that A∣(a,b) ∈ O(Ω,Mn(L1(a, b))).

It then follows from Lemma 89 with p = ∞, q = s = 1 that we have A := J −1A∣(a,b) ∈

O(Ω,Mn(L1(a, b))). But according to our definition we have

T ∣(a,b)(!) = ′ − A(⋅, !) , ∈ (W 1,1(a, b))n, ! ∈ Ω.

with A ∈ O(Ω,Mn(L1(a, b))). It now follows from the fact ()′ ∈ L ((W 1,1(a, b))n, (L1(a, b))n)

and Lemma 83 that T ∣(a,b)(!) : (W 1,1(a, b))n → (L1(a, b))n is a well-defined linear operator

and T ∣(a,b)(!) ∈ L ((W 1,1(a, b))n, (L1(a, b))n), for each ! ∈ Ω. It follows now from both

Lemma 79 and Lemma 90 that T ∣(a,b) ∈ O(Ω,L ((W 1,1(a, b))n, (L1(a, b))n)).

Now let ! ∈ Ω, ∈ (W 1,1loc (ℝ))n, and (a, b) ⊆ ℝ) be a bounded interval. Then by Lemma

69 we have ∣(a,b) ∈ (W 1,1(a, b))n and so T ∣(a,b)(!) ∣(a,b) ∈ (L1(a, b))n. But (T (!) )∣(a,b) ∈

(L1(a, b))n since T (!) ∈ (L1loc(ℝ))n and by Lemma 69 we have ∣′(a,b) = ′∣(a,b). Thus we

conclude that

(T (!) )∣(a,b) = ( ′ −J −1A(⋅, !) )∣(a,b) = ∣′(a,b) −J −1A(⋅, !)∣(a,b) ∣(a,b)

112

= T ∣(a,b)(!) ∣(a,b).

And hence we have shown that (T (!) )∣(a,b) = T ∣(a,b)(!) ∣(a,b) for every ! ∈ Ω, every

∈ (W 1,1loc (ℝ))n, and every bounded interval (a, b) ⊆ ℝ. This completes the proof.

Corollary 54 Let ! ∈ Ω. Then the following statements are equivalent:

(i) is a solution of the canonical ODEs in (4.13) with frequency !.

(ii) ∈ ker(T (!)).

(iii) ∈ (W 1,1loc (ℝ))n and ∣(a,b) ∈ ker(T ∣(a,b)(!)) every bounded interval (a, b) ⊆ ℝ.

Proof. We have already proven in Lemma 52 that statement (i) is equivalent to statement

(ii). We now complete the proof of this corollary by proving statement (ii) is equivalent to

statement (iii). Suppose ∈ ker(T (!)) for some ! ∈ Ω. Then ∈ (W 1,1loc (ℝ))n and for any

bounded interval (a, b) ⊆ ℝ by Proposition 53 we have 0 = (T (!) )∣(a,b) = T ∣(a,b)(!) ∣(a,b) im-

plying ∣(a,b) ∈ ker(T ∣(a,b)(!)). Conversely, if ∈ (W 1,1loc (ℝ))n and ∣(a,b) ∈ ker(T ∣(a,b)(!)) for

every bounded interval (a, b) ⊆ ℝ then by Proposition 53 this implies 0 = T ∣(a,b)(!) ∣(a,b) =

(T (!) )∣(a,b) for every bounded interval (a, b) ⊆ ℝ. But since T (!) ∈ (L1loc(ℝ))n this

implies T (!) = 0 in (L1loc(ℝ))n and therefore ∈ ker(T (!)). This completes the proof.

In the following proposition, we collect together some facts from [45, §II.2.5] that will be

need in this chapter.

Proposition 55 For every bounded interval (a, b) ⊆ ℝ with 0 ∈ (a, b) and for each ! ∈ Ω,

there exists a unique function Ψ∣(a,b)(⋅, !) ∈ Mn(W 1,1(a, b)) satisfying for a.e. x ∈ (a, b) the

113

matrix differential equation with initial condition

Ψ′(x) = J −1A(x, !)Ψ(x), Ψ(0) = In. (4.53)

Moreover, the following statements are true:

(i) If ∈ ker(T ∣(a,b)(!)) then there exists a unique ∈ ℂn such that = Ψ∣(a,b)(⋅, !) .

Conversely, if ∈ ℂn and we define := Ψ∣(a,b)(⋅, !) then ∈ ker(T ∣(a,b)(!)).

(ii) For every (x, !) ∈ [a, b]× Ω,

Ψ∣(a,b)(x, !) = In +

∫ x

0

J −1A(t, !)Ψ∣(a,b)(t, !)dt. (4.54)

(iii) As a function of frequency, Ψ∣(a,b) ∈ O(Ω,Mn(W 1,1(a, b))).

(iv) For every (x, !) ∈ [a, b] × Ω, Ψ∣−1(a,b)(x, !) := Ψ∣(a,b)(x, !)−1 exists and as function of

position Ψ∣−1(a,b)(⋅, !) ∈Mn(W 1,1(a, b)).

Proof. Let us fix a bounded interval (a, b) ⊆ ℝ containing 0. We proceed by proving the

existence and uniqueness portions of this proposition first and then we will prove statements

(i)–(iv).

We begin by reminding the reader that, by the proof of Proposition 53, we have

T ∣(a,b)(!) = ′ − A(⋅, !) , ∈ (W 1,1(a, b))n, ! ∈ Ω.

with A ∈ O(Ω,Mn(L1(a, b))) where A(⋅, !) := J −1A(⋅, !)∣(a,b) for ! ∈ Ω. Following [45, p.

69, §II.2.5, Definition 2.5.2], we introduce two new definitions.

Definition 25 Let !0 ∈ Ω. A matrix Y0 ∈Mn(W 1,1(a, b)) is called a fundamental matrix

of T ∣(a,b)(!0) = 0 if for each ∈ ker(T ∣(a,b)(!0)) there exists a ∈ ℂn such that = Y0 .

114

A matrix function Y : Ω→ Mn(W 1,1(a, b)) is called a fundamental matrix function of

T ∣(a,b) = 0 if Y (!) is a fundamental matrix of T ∣(a,b)(!) for each ! ∈ Ω.

We now prove the existence portion of this proposition. By [45, p. 69, §II.2.5, Theorem 2.5.3]

there exists a fundamental matrix function Y ∈ O(Ω,Mn(W 1,1(a, b))) of T ∣(a,b) = 0 with

the property that for all ! ∈ Ω, Y (a, !) = In and Y (⋅, !) is invertible in Mn(W 1,1(a, b)).

Hence for each ! ∈ Ω, there exists Y −1(⋅, !) ∈ Mn(W 1,1(a, b)) such that Y −1(⋅, !)Y (⋅, !) =

Y (⋅, !)Y −1(⋅, !) = In. But the functions Y (⋅, !), Y −1(⋅, !) : [a, b] → Mn(ℂ) are continuous

implying Y (x, !)Y −1(x, !) = In for all x ∈ [a, b]. Thus Y (x, !) is invertible for all x ∈ [a, b]

and Y −1(x, !) = Y (x, !)−1.

Now for each ! ∈ Ω, we define Ψ∣(a,b)(⋅, !) := Y (⋅, !)Y −1(0, !). It follows by [45, p. 71,

§II.2.5, Proposition 2.5.4] that Ψ∣(a,b)(⋅, !) ∈ Mn(W 1,1(a, b)) is a fundamental matrix of

T ∣(a,b)(!) = 0 for each ! ∈ Ω. It then follows from [45, p. 72, §II.2.5, Corollary 2.5.5] that

Ψ∣(a,b)(⋅, !) is a solution of the equation

Ψ′ − A(⋅, !)Ψ = 0

in Mn(W 1,1(a, b)) for each ! ∈ Ω. But this means for each ! ∈ Ω, Ψ∣(a,b)(⋅, !) satisfies for

a.e. x ∈ (a, b) the matrix differential equation with initial condition

Ψ′(x) = J −1A(x, !)Ψ(x), Ψ(0) = In. (4.55)

This proves existence.

We now prove the uniqueness portion of this proposition. Let ! ∈ Ω. Suppose Φ ∈

Mn(W 1,1(a, b)) satisfies for a.e. x ∈ (a, b) the matrix differential equation with initial condi-

115

tion

Ψ′(x) = J −1A(x, !)Ψ(x), Ψ(0) = In. (4.56)

It follows that Φ is a solution of the equation

Ψ′ − A(⋅, !)Ψ = 0

in Mn(W 1,1(a, b)) with 0 ∈ [a, b] and Φ(0) = In. This implies by [45, p. 73, §II.2.5, Propo-

sition 2.5.9] that Φ is a fundamental matrix of T ∣(a,b)(!) = 0. But from the existence

portion of this proof, Y (⋅, !) is also a fundamental matrix of T ∣(a,b)(!) = 0 and so by [45,

p. 71, §II.2.5, Proposition 2.5.4] there exists an invertible matrix C ∈Mn(ℂ) such that Φ =

Y (⋅, !)C. Thus since In = Φ(0) = Y (0, !)C this implies Φ = Y (⋅, !)Y −1(0, !) = Ψ∣(a,b)(⋅, !).

This proves uniqueness.

Now we will prove statements (i)–(iv) of this proposition. We begin with statement (i). First,

it follows from what we just proved that for each ! ∈ Ω, Ψ∣(a,b)(⋅, !) = Y (⋅, !)Y −1(0, !) ∈

Mn(W 1,1(a, b)) is a fundamental matrix of T ∣(a,b)(!) = 0. By Definition 25 this means

if ∈ ker(T ∣(a,b)(!)) then there exists ∈ (ℂ)n such that = Ψ∣(a,b)(⋅, !) . But as

discussed Y (⋅, !) is invertible with inverse Y −1(⋅, !)−1 ∈ Mn(W 1,1(a, b)). Uniqueness of the

representation = Ψ∣(a,b)(⋅, !) follows since = Y (0, !)Y −1(⋅, !) in (W 1,1(a, b))n.

Let ! ∈ Ω, let ∈ (ℂ)n, and define := Ψ∣(a,b)(⋅, !) . Now because = Y (⋅, !)(Y −1(0, !) )

and Y is a fundamental matrix function of T ∣(a,b) = 0, it follows by Definition 25 that

∈ ker(T ∣(a,b)(!)). This completes the proof of statement (i).

116

The proof of statement (ii) is straightforward. Let (x, !) ∈ [a, b]×Ω. Then Ψ := Ψ∣(a,b)(⋅, !)

satisfies (4.53) and so upon integrating and applying Lemma 70 we arrive at

Ψ∣(a,b)(x, !) = In +

∫ x

0

Ψ′(t, !)dt = In +

∫ x

0

J −1A(t, !)Ψ(t, !)dt

= In +

∫ x

0

J −1A(t, !)Ψ∣(a,b)(⋅, !)dt.

This proves statement (ii).

We complete the proof of this proposition now by proving statements (iii) and (iv). We have

already shown that there exists Y ∈ O(Ω,Mn(W 1,1(a, b))) such that for each ! ∈ Ω, Y (⋅, !)

is invertible in Mn(W 1,1(a, b)) and, denoting this inverse in Mn(W 1,1(a, b)) by Y −1(⋅, !), we

have for every x ∈ [a, b], Y (x, !) is invertible with Y −1(x, !) = Y (x, !)−1 and Ψ∣(a,b)(⋅, !) =

Y (⋅, !)Y −1(0, !). This proves statement (iv) since for every (x, !) ∈ [a, b] × Ω, we have

Ψ∣(a,b)(x, !) is invertible with inverse Ψ∣−1(a,b)(x, !) := Ψ∣(a,b)(x, !)−1 = Y (0, !)Y −1(x, !) and

Ψ∣−1(a,b)(⋅, !) = Y (0, !)Y −1(⋅, !) ∈Mn(W 1,1(a, b)) by Lemma 72.

We will now show Ψ∣(a,b) ∈ O(Ω,Mn(W 1,1(a, b))). Since Y ∈ O(Ω,Mn(W 1,1(a, b))) and

Y (⋅, !) is invertible in Mn(W 1,1(a, b)) with the inverse Y −1(⋅, !) for every ! ∈ Ω, it follows

from [45, p. 66, §II.2.3, Proposition 2.3.3] and [45, p. 7, §I.1.2, Proposition 1.2.5] that

Y −1 ∈ O(Ω,Mn(W 1,1(a, b))). It follows from this and Lemma 91 that if we let �0 denote the

evaluation map at x = 0 then Y −1(0, ⋅) = �0Y−1 ∈ O(Ω,Mn(ℂ)). But the map � : Mn(ℂ)→

Mn(W 1,1(a, b)) given by (�B)(⋅) := B is a continuous linear map and so by Lemma 78 we

have Y −1(0, ⋅) = ��0Y−1 ∈ O(Ω,Mn(W 1,1(a, b))). Therefore by Lemma 78 and Lemma 72

we have Ψ∣(a,b) = Y −1��0Y−1 ∈ O(Ω,Mn(W 1,1(a, b))). This proves statement (iii) and hence

the proof of the proposition is complete.

From the functions Ψ∣(a,b)(⋅, !) we define a matrix-valued function Ψ : ℝ× Ω→Mn(ℂ) by

Ψ(x, !) := Ψ∣(a,b)(x, !) (4.57)

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for each (x, !) ∈ ℝ× Ω and for any interval (a, b) containing 0 and x.

Lemma 56 The matrix-valued function Ψ : ℝ× Ω→Mn(ℂ) is well-defined. Moreover, for

each ! ∈ Ω, Ψ(⋅, !) ∈Mn(W 1,1loc (ℝ)) and for any a, b ∈ ℝ with a < 0 < b we have

Ψ(⋅, !)∣(a,b) = Ψ∣(a,b)(⋅, !). (4.58)

Proof. First, fix ! ∈ Ω and let (u, v) be an interval such that 0 ∈ (u, v) ⊆ (a, b). We will

begin by showing

Ψ∣(a,b)(⋅, !)∣(u,v) = Ψ∣(u,v)(⋅, !). (4.59)

We define Y := Ψ∣(a,b)(⋅, !)∣(u,v) so that Y (0) = In. Then by [45, §II.2.2, Proposition 2.2.1]

we have Y ∈ Mn(W 1,1(u, v)) and Y ′ = Ψ∣(a,b)(⋅, !)′∣(u,v). This implies by the definition of

Ψ∣(a,b)(⋅, !) in Proposition 55 that Y is a function in Mn(W 1,1(u, v)) satisfying the for a.e.

x ∈ (u, v) the matrix differential equation with initial condition

Ψ′(x) = J −1A(x, !)Ψ(x), Ψ(0) = In.

By the uniqueness portion of Proposition 55 we conclude that Y := Ψ∣(u,v)(⋅, !). Thus we

have shown that Ψ∣(a,b)(⋅, !)∣(u,v) = Ψ∣(u,v)(⋅, !).

We now prove the function Ψ defined by (4.57) is well-defined. Let (x, !) ∈ ℝ × Ω. Let

(a1, b1) and (a2, b2) be any intervals containing 0 and x. Then there exists u, v such that

(u, v) = (a1, b1) ∩ (a2, b2). Now since 0 and x are in the interval (u, v) then Ψ∣(u,v)(⋅, !)

and Ψ∣(u,v)(x, !) are well-defined by Proposition 55. And thus since 0 ∈ (u, v) ⊆ (aj, bj) for

j = 1, 2, we get from (4.59) that

Ψ∣(a1,b1)(x, !) = Ψ∣(u,v)(x, !) = Ψ∣(a2,b2)(x, !).

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This proves the function Ψ is well-defined.

Next, to show that Ψ(⋅, !) ∈Mn(W 1,1loc (ℝ)) for any ! ∈ Ω its enough to show, by Lemma 71,

that Ψ(⋅, !)∣(a,b) ∈ Mn(W 1,1(a, b)) for any interval (a, b) containing 0. But this follows from

the definition (4.57) since for any x ∈ (a, b)

Ψ(x, !) = Ψ∣(a,b)(x, !)

which implies

Ψ(⋅, !)∣(a,b) = Ψ∣(a,b)(⋅, !) ∈Mn(W 1,1(a, b)).

This completes the proof of the lemma.

We are in a position now to prove Proposition 33.

Proof. [Proposition 33] To begin this proof we need to show that the matrix-valued

function Ψ : ℝ×Ω→Mn(ℂ), as defined by (4.57), for a fixed ! ∈ Ω satisfies a.e. x ∈ ℝ the

matrix differential equation with initial condition

J Ψ′(x) = A(x, !)Ψ(x), Ψ(0) = In,

and Ψ(⋅, !) is the only function in Mn(W 1,1loc (ℝ)) that does so. After which we will prove that

it has the properties described in Proposition 33.

This first part follows now from the previous lemma since for any ! ∈ Ω and interval (a, b)

containing 0 we have Ψ(x, !)∣(a,b) = Ψ∣(a,b)(x, !) for all x ∈ (a, b) and so by Proposition

55 the function Ψ(⋅, !) satisfies a.e. x ∈ (a, b) the matrix differential equation with initial

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condition

Ψ′(x) = J −1A(x, !)Ψ(x), Ψ(0) = In.

Hence this implies Ψ(⋅, !) satisfies a.e. x ∈ ℝ the matrix differential equation with initial

condition

J Ψ′(x) = A(x, !)Ψ(x), Ψ(0) = In.

Lets now prove uniqueness. Fix ! ∈ Ω. Suppose Ψ1 ∈Mn(W 1,1loc (ℝ)) satisfied a.e. x ∈ ℝ the

matrix differential equation with initial condition

J Ψ′(x) = A(x, !)Ψ(x), Ψ(0) = In.

Then for any interval (a, b) containing 0, Ψ1∣(a,b) ∈Mn(W 1,1(a, b)) and satisfies a.e. x ∈ (a, b)

the matrix differential equation with initial condition

Ψ′(x) = J −1A(x, !)Ψ(x), Ψ(0) = In.

and so by the uniqueness part of Proposition 55 this implies Ψ1(⋅)∣(a,b) = Ψ∣(a,b)(⋅, !). By

definition (4.57) this implies Ψ1(⋅)∣(a,b) = Ψ(⋅, !)∣(a,b). Since this is true for any interval (a, b)

containing 0 we must have Ψ1(⋅) = Ψ(⋅, !). This proves uniqueness.

Now we will prove Propositions 33.(i)–(vi). We start with property (i). Suppose that is

a solution of the canonical ODEs in (4.13) at the frequency ! ∈ Ω. Then we must prove

there exists a unique ∈ ℂn such that, as elements of (W 1,1loc (ℝ))n, = Ψ(⋅, !) . We begin

by proving the uniqueness statement. Suppose that there did exist ∈ ℂn such that =

Ψ(⋅, !) . It follows from this, since ∈ (W 1,1loc (ℝ))n and Ψ(⋅, !) ∈ Mn(W 1,1

loc (ℝ)) by Lemma

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70 they have unique locally absolutely continuous representative functions (⋅) : ℝ → ℂn

and Ψ(⋅, !) : ℝ → Mn(ℂ) which are continuous, that (x) = Ψ(x, !) for all x ∈ ℝ. In

particular, since Ψ(0, !) = In this means (0) = Ψ(0, !) = . This proves uniqueness of .

Lets now prove existence. By Lemma 52 we have ∈ ker(T (!)). By Corollary 54 we have

∣(a,b) ∈ ker(T ∣(a,b)(!)) for any bounded interval (a, b) ⊆ ℝ containing 0 and so by Proposition

55.((i)) there exists a ∣(a,b) ∈ ℂn such that ∣(a,b) = Ψ∣(a,b)(⋅, !) ∣(a,b). But 0 ∈ (a, b) and so

by (4.57) we have := (0) = (0)∣(a,b) = ∣(a,b)(0) = Ψ∣(a,b)(0, !) ∣(a,b) = Ψ(0, !) ∣(a,b) =

∣(a,b). Thus from this and Lemma 56 we have ∣(a,b) = Ψ∣(a,b)(⋅, !) = Ψ(⋅, !)∣(a,b) =

Ψ(⋅, !) ∣(a,b). But since this is true for every bounded interval (a, b) ⊆ ℝ containing 0 this

implies as elements of (W 1,1loc (ℝ))n we have = Ψ(⋅, !) . This proves existence.

Conversely, let ∈ (ℝ)n and define := Ψ(⋅, !) . By Lemma 73 we have ∈ (W 1,1loc (ℝ))n.

Then for any bounded interval (a, b) ⊆ ℝ containing 0 we have by Lemma 56 and Proposition

55.(i) that ∣(a,b) = Ψ(⋅, !)∣(a,b) = Ψ∣(a,b)(⋅, !) ∈ ker(T ∣(a,b)(!)). It follows from this and

Proposition 53 that we have (T (!) )∣(a,b) = T ∣(a,b)(!) ∣(a,b) = 0 in (L1(a, b))n. But this is

true for every bounded interval (a, b) ⊆ ℝ containing 0 and so it follows this and Lemma 4.47

that T (!) = 0 in (L1loc(ℝ))n which means ∈ ker(T (!)). Therefore by Corollary 54 this

implies is a solution of the canonical ODEs in (4.13) with frequency !. This completes

the proof of property (i).

Next we will prove property (ii). Let (x, !) ∈ ℝ × Ω. Take any interval (a, b) containing 0

and x. Then by Proposition 55 and the previous lemma we have

Ψ(x, !) = Ψ∣(a,b)(x, !) = In +

∫ x

0

J −1A(t, !)Ψ∣(a,b)(t, !)dt

= In +

∫ x

0

J −1A(t, !)Ψ(t, !)dt.

This proves property (ii).

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Next we will prove property (iii). That is, we must show for each fixed x ∈ ℝ, that Ψ(x, ⋅) ∈

O(Ω,Mn(ℂ)). To do this we fix x ∈ ℝ and choose any bounded interval (a, b) ⊆ ℝ with

0, x ∈ (a, b). Then by (4.57) we have Ψ(x, ⋅) = Ψ∣(a,b)(x, ⋅). Let �x : Mn(W 1,1(a, b))→Mn(ℂ)

be the evaluation map at x defined by �xB := B(x) – the value of the unique absolutely

continuous representative of B ∈ Mn(W 1,1(a, b)) at x. Then by Proposition 55.(iii) we

have Ψ∣(a,b) ∈ O(Ω,Mn(W 1,1(a, b))) and so by Lemma 91 we have �xΨ∣(a,b) ∈ O(Ω,Mn(ℂ)).

And thus we have Ψ(x, ⋅) = Ψ∣(a,b)(x, ⋅) = �xΨ∣(a,b) ∈ O(Ω,Mn(ℂ)) as desired. This proves

property (iii).

Next we prove property (iv). To start, since Ψ(x, ⋅) ∈ O(Ω,Mn(ℂ)) for each x ∈ ℝ, we can

define Ψ! : ℝ × Ω → Mn(ℂ) to be the partial derivative with respect to frequency in the

Mn(ℂ) norm of the function Ψ : ℝ× Ω→Mn(ℂ), i.e.,

Ψ!(x, !0) := lim!→!0

(! − !0)−1(Ψ(x, !)−Ψ(x, !0)), ∀(x, !0) ∈ ℝ× Ω.

Our goal is to prove that for any (x, !0) ∈ ℝ× Ω, Ψ!(⋅, !0) ∈Mn(W 1,1loc (ℝ)) and

Ψ!(x, !0) =

∫ x

0

J −1A!(t, !0)Ψ(t, !0) + J −1A(t, !0)Ψ!(t, !0)dt.

To do this we begin by choosing any bounded interval (a, b) ⊆ ℝ with 0 ∈ (a, b). Next, let

ℐ : Mn(L1(a, b))→Mn(W 1,1(a, b)) be the integral map defined by

(ℐB)(x) :=

∫ x

0

B(t)dt, B ∈Mn(L1(a, b)), x ∈ (a, b).

Now by hypothesis we have A ∈ O(Ω, L1(T)) and so by Lemma 93 we have A∣(a,b) ∈

O(Ω, L1(a, b)) and (A∣(a,b))! = A!∣(a,b). It then follows from this and Lemma 89 with

p = ∞, q = s = 1 that J −1A∣(a,b) ∈ O(Ω, L1(a, b)) and (J −1A∣(a,b))! = J −1A!∣(a,b).

From this and the fact by Proposition 55.(iii) we have Ψ∣(a,b) ∈ O(Ω,Mn(W 1,1(a, b))), it fol-

lows by Lemma 90 with p = 1 that J −1A∣(a,b)Ψ∣(a,b) ∈ O(Ω,Mn(L1(a, b))) whose derivative

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is (J −1A∣(a,b)Ψ∣(a,b))! = J −1A!∣(a,b)Ψ∣(a,b) + J −1A∣(a,b)(Ψ∣(a,b))!. It follows from this and

Lemma 92 that we must have ℐ(J −1A∣(a,b)Ψ∣(a,b)) ∈ O(Ω,Mn(W 1,1(a, b))) and

(ℐ(J −1A∣(a,b)Ψ∣(a,b)))! = ℐ(J −1A!∣(a,b)Ψ∣(a,b) + J −1A∣(a,b)(Ψ∣(a,b))!).

This and Lemma 91 imply that for any x ∈ (a, b), �xℐ(J −1A∣(a,b)Ψ∣(a,b)) ∈ O(Ω,Mn(ℂ))

and

(�xℐ(J −1A∣(a,b)Ψ∣(a,b)))! = �xℐ(J −1A!∣(a,b)Ψ∣(a,b) + J −1A∣(a,b)(Ψ∣(a,b))!). (4.60)

Now from Proposition 55.(ii) it follows that we have the identity

Ψ(x, ⋅) = Ψ∣(a,b)(x, ⋅) = �xΨ∣(a,b) = In + �xℐ(J −1A∣(a,b)Ψ∣(a,b)), ∀x ∈ (a, b).

From this and (4.60) it follows that

Ψ!(x, ⋅) = �x(Ψ∣(a,b))! = (In + �xℐ(J −1A∣(a,b)Ψ∣(a,b)))!

= �xℐ(J −1A!∣(a,b)Ψ∣(a,b) + J −1A∣(a,b)(Ψ∣(a,b))!), ∀x ∈ (a, b). (4.61)

But this implies for any (x, !0) ∈ (a, b) × Ω we have Ψ!(x, !0) = �x(Ψ∣(a,b))!(!0) =

(Ψ∣(a,b))!(x, !0). And hence this shows that for each !0 ∈ Ω,

Ψ!(⋅, !0)∣(a,b) = (Ψ∣(a,b))!(⋅, !0) ∈Mn(W 1,1(a, b)). (4.62)

But since this is true for any bounded interval (a, b) ⊆ ℝ with 0 ∈ (a, b), then by Lemma

71 for any !0 ∈ Ω we have Ψ!(⋅, !0) ∈ Mn(W 1,1loc (ℝ)). Moreover, it follows that for any

(x, !0) ∈ ℝ × Ω we can choose a bounded interval (a, b) ⊆ ℝ with 0, x ∈ (a, b) so that by

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(4.57), (4.61), and (4.62) we have

Ψ!(x, !0) = �xℐ(J −1A!∣(a,b)Ψ∣(a,b) + J −1A∣(a,b)(Ψ∣(a,b))!)(!0)

=

∫ x

0

J −1A!∣(a,b)(t, !0)Ψ∣(a,b)(t, !0) + J −1A∣(a,b)(t, !0)(Ψ∣(a,b))!(t, !0)dt

=

∫ x

0

J −1A!(t, !0)Ψ(t, !0) + J −1A(t, !0)Ψ!(t, !0)dt.

This completes the proof of Proposition 33.(iv).

Next we prove property (v) of Proposition 33. The goal is to prove for every (x, !) ∈ ℝ×Ω

the matrix Ψ(x, !) is invertible and, denoting Ψ−1(x, !) := Ψ(x, !)−1, that Ψ−1(⋅, !) ∈

Mn(W 1,1loc (ℝ)). Well, by Proposition 55.(iv) for any bounded interval (a, b) ⊆ ℝ containing

0 and any (x, !) ∈ (a, b)× Ω we have Ψ∣(a,b)(x, !) is invertible and denoting Ψ∣−1(a,b)(x, !) :=

Ψ∣(a,b)(x, !)−1 we have Ψ∣−1(a,b)(⋅, !) ∈ Mn(W 1,1(a, b)). But by (4.57) and since Ψ(x, !) =

Ψ∣(a,b)(x, !) this implies Ψ−1(x, !) = Ψ∣−1(a,b)(x, !) for any (x, !) ∈ (a, b)×Ω. This proves for

every (x, !) ∈ ℝ × Ω the matrix Ψ(x, !) is invertible and for any ! ∈ Ω, Ψ−1(⋅, !)∣(a,b) =

Ψ∣−1(a,b)(⋅, !) ∈Mn(W 1,1(a, b)) for any bounded interval (a, b) ⊆ ℝ containing 0. And therefore

by Lemma 71 this proves Ψ−1(⋅, !) ∈Mn(W 1,1loc (ℝ)) for any ! ∈ Ω. This completes the proof

of property (v) of Proposition 33.

Finally, we will complete the proof of Proposition 33 by proving property (vi). To do this

we must show that for every (x, !) ∈ ℝ × Ω we have Ψ(x + d, !) = Ψ(x, !)Ψ(d, !). Fix

! ∈ Ω and define Φ := Ψ(⋅ + d, !)Ψ(d, !)−1. We now show that Φ ∈ Mn(W 1,1loc (ℝ)). First,

since Ψ(⋅, !) ∈ Mn(W 1,1loc (ℝ)), it follows from Lemma 74 that Ψ(⋅ + d, !) ∈ Mn(W 1,1

loc (ℝ)).

Second, since Ψ(d, !)−1 is just a constant matrix then Ψ(d, !)−1 ∈ Mn(W 1,1loc (ℝ)). It now

follows from Lemma 73 that the product Φ = Ψ(⋅+ d, !)Ψ(d, !)−1 ∈Mn(W 1,1loc (ℝ)). We will

now show that Φ ∈ Mn(W 1,1loc (ℝ)) satisfies a.e. the matrix differential equation with initial

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condition

J Ψ′(x) = A(x, !)Ψ(x), Ψ ∈Mn(W 1,1loc (ℝ)), Ψ(0) = In.

To do this we let ℒd : Mn(W 1,1loc (ℝ)) → Mn(W 1,1

loc (ℝ)) denote the translation operator as

defined in Lemma 74. Then Φ = ℒd(Ψ(⋅, !)Ψ(d, !)−1). Then by Lemma 74 and Lemma 73

we have

J Φ′ = J (ℒd(Ψ(⋅, !)Ψ(d, !)−1))′ = Jℒd(Ψ(⋅, !)Ψ(d, !)−1)′

= Jℒd(Ψ(⋅, !)′Ψ(d, !)−1) = ℒd(J Ψ(⋅, !)′)Ψ(d, !)−1.

But Proposition 33 implies J Ψ(⋅, !)′ = A(⋅, !)Ψ(⋅, !) and since by hypothesis A(⋅+d, !) =

A(⋅, !) this implies ℒd(A(⋅, !)Ψ(⋅, !)) = A(⋅ + d, !)Ψ(⋅ + d, !) = A(⋅, !)ℒd(Ψ(⋅, !)). Hence

these facts imply

J Φ′ = ℒd(J Ψ(⋅, !)′)Ψ(d, !)−1 = ℒd(A(⋅, !)Ψ(⋅, !))Ψ(d, !)−1

= A(⋅, !)ℒd(Ψ(⋅, !))Ψ(d, !)−1 = A(⋅, !)Φ.

But this is an equality of elements in Mn(L1loc(ℝ)) and so together with the fact that Φ(0) =

Ψ(0 + d, !)Ψ(d, !)−1 = In this implies that Φ ∈ Mn(W 1,1loc (ℝ)) satisfies a.e. the matrix

differential equation with initial condition

J Ψ′(x) = A(x, !)Ψ(x), Ψ ∈Mn(W 1,1loc (ℝ)), Ψ(0) = In.

Now though the uniqueness portion of Proposition 33 implies that we must have Ψ(⋅, !) = Φ.

Thus Ψ(x + d, !) = Ψ(x, !)Ψ(d, !) for a.e. x ∈ ℝ and so, since Proposition 33.(ii) implies

the function Ψ(x, !) is continuous as a function of x ∈ ℝ, we must have Ψ(x + d, !) =

Ψ(x, !)Ψ(d, !) for all x ∈ ℝ. This proves Proposition 33.(vi) and therefore completes the

125

proof of Proposition 33.

We will use Proposition 33 now to prove the Floquet-Lyapunov Theorem.

Proof. [Theorem 34] We fix ! ∈ Ω. To simplify notation we will denote the matricant

by Φ := Ψ(⋅, !) and we will agree to treat, in the rest of this proof, all objects as depending

implicitly on the frequency !. Our goal here is to show that there exists a matrix K ∈Mn(ℂ)

and a function F ∈Mn(W 1,1loc (ℝ)) such that for every x ∈ ℝ,

Φ(x) = F (x)eixK ,

where F (x+ d) = F (x), F (0) = In, F−1(x) := F (x)−1 exist, and F−1 ∈Mn(W 1,1loc (ℝ)).

To begin, let �1, . . . , �s denote the distinct eigenvalues of the monodromy matrix Φ(d). Let

log(z) be any branch of the logarithm which is analytic at each �j, for j = 1, . . . , s. Using

the Riesz-Dunford functional calculus as described for matrices in [35, pp. 304–334, §9], it

follows that the matrix K := 1id

log(Φ(d)) satisfies

eidK = Φ(d),

where ez is the exponential function. From the functional calculus it follows that the matrix

function B(x) := eixK , for x ∈ ℝ has the properties B ∈ Mn(W 1,1loc (ℝ)), B(x) is invertible

with inverse B−1(x) := e−ixK for x ∈ ℝ, and B−1 ∈Mn(W 1,1loc (ℝ)).

We define the function F by F (x) := Φ(x)e−ixK , x ∈ ℝ. It then follows from Lemma

73 that F ∈ Mn(W 1,1loc (ℝ)). We also find that for every x ∈ ℝ, F (x) is invertible with

inverse F−1(x) := eixKΦ(x)−1. It follows from Proposition 33.(v) and Lemma 73 that F−1 ∈

Mn(W 1,1loc (ℝ)). Furthermore, since Φ(0) = In = e−i0K then F (0) = In. Moreover, since

eidK = Φ(d), e−idK = Φ(d)−1, and, by Proposition 33.(vi), Φ(x + d) = Φ(x)Φ(d) for every

126

x ∈ ℝ, we have

F (x+ d) = Φ(x+ d)e−i(x+d)K = Φ(x)Φ(d)e−idKe−ixK = Φ(x)Φ(d)Φ(d)−1e−ixK

= Φ(x)e−ixK = F (x), ∀x ∈ ℝ.

This completes the proof of Theorem 34.

Using the Floquet-Lyapunov Theorem and its proof we will now establish the validity of

Theorem 35.

Proof. [Theorem 35] We first prove the statement: if is a nontrivial Floquet solution

of the canonical ODEs in (4.13) with wavenumber-frequency pair (k, !), Floquet multiplier

� = eikd, and order m ∈ ℕ ∪ {0} then = Ψ(⋅, !) where is a generalized eigenvector of

the monodromy matrix Ψ(d, !) of order m+ 1 corresponding to the eigenvalue �. We prove

this by induction on the order m of the Floquet solution. Before we begin we will prove the

following lemma:

Lemma 57 If {uj}mj=0 ⊆ (W 1,1loc (ℝ))n ∩ (L1(T))n and for a.e. x ∈ ℝ,

m∑j=0

xjuj(x) = 0

then u1 = ⋅ ⋅ ⋅ = um = 0.

Proof. We prove the statement by induction. The statement is obviously true for m = 0.

Suppose the statement is true for m ∈∈ ℕ ∪ {0}. Lets show its true for m + 1. Suppose

{uj}m+1j=0 ⊆ (W 1,1

loc (ℝ))n ∩ (L1(T))n and satisfies for a.e. x ∈ ℝ,

m+1∑j=0

xjuj(x) = 0

127

By Lemma 70 we can assume {uj}m+1j=0 ⊆ (ACloc(ℝ))n and satisfy uj(x + d) = uj(x) for all

x ∈ ℝ and for j = 0, . . . ,m + 1. Hence letting uj,i denote the ith row entry of uj we have

uj,i ∈ ACloc(ℝ) is a d-periodic function and hence is bounded on ℝ, for j = 0, . . . ,m + 1,

i = 0, . . . , n. This implies

limx→∞

xjuj,i(x)

xm+1= 0

for j = 0, . . . ,m, i = 0, . . . , n and there exists xi ∈ [0, d) such that

um+1,i(xi) = supx∈ℝ∣um+1,i(x)∣

for i = 0, . . . , n. Now it follows from our hypothesis that

m+1∑j=0

xjuj,i(x) = 0, ∀x ∈ ℝ, i = 1, . . . , n.

Thus we conclude for l ∈ ℕ,

supx∈ℝ∣um+1,i(x)∣ = um+1,i(xi) = lim

l→∞um+1,i(xi + ld)

= liml→∞−

m∑j=0

(xi + ld)juj,i(xi + ld)

(xi + ld)m+1= 0

for i = 0, . . . , n. This implies um+1 = 0. But now the sequence {uj}mj=0 satisfies the hypothe-

ses of this lemma and so by the induction hypotheses we conclude that um = um−1 = ⋅ ⋅ ⋅ =

u0 = 0. Therefore by induction the statement is true for all m ∈ ℕ ∪ {0}. This completes

the proof.

We now begin the proof of the above statement. Let us show the base case m = 0 is

true. Suppose is a Floquet solution of the canonical ODEs in (4.13) with wavenumber-

frequency pair (k, !), Floquet multiplier � = eikd, and order m = 0. Then by Definition

128

19, (x) = eikxu0(x) for a.e. x ∈ ℝ, for some u0 ∈ (W 1,1loc (ℝ))n ∩ (L1(T))n with u0 ∕= 0. By

Lemma 57 we conclude ∕= 0 and so by Proposition 33.(i), = Ψ(⋅, !) for some nonzero

∈ ℂn. These facts and Proposition 33.(vi) imply for a.e. x ∈ ℝ, Ψ(x, !)Ψ(d, !) =

(x + d) = eikxeikdu0(x + d) = �eikdu0(x) = � (x) = Ψ(x, !)� . Hence Proposition 33.(v)

implies Ψ(d, !) = � . But this implies, since ∕= 0, that is a generalized eigenvector of

Ψ(d, !) of order 1 corresponding to the eigenvalue �. This proves the base case.

Suppose now the statement is true for m ∈ ℕ ∪ {0}. We will now show its true for m + 1.

Let be a Floquet solution of the canonical ODEs in (4.13) with wavenumber-frequency

pair (k, !), Floquet multiplier � = eikd, and order m + 1. Then by Definition 19, (x) =

eikx∑m+1

j=0 xjuj(x) for a.e. x ∈ ℝ, for some {uj}m+1j=0 ⊆ (W 1,1

loc (ℝ))n ∩ (L1(T))n with um+1 ∕= 0.

By Lemma 57 we conclude ∕= 0 and so by Proposition 33.(i), = Ψ(⋅, !) for some nonzero

∈ ℂn. We define

:= (Ψ(d, !)− �In) , := Ψ(⋅, !) .

We will now prove is a Floquet solution of the canonical ODEs in (4.13) with wavenumber-

frequency pair (k, !), Floquet multiplier � = eikd, and order m. First, it follows by Proposi-

tion 33.(i) that is a solution of the canonical ODEs in (4.13) at the frequency ! ∈ Ω. Next,

by Proposition 33.(vi), the Floquet representation of , and the fact (x+d)j =∑j

l=0

(jl

)dj−lxl,

it follows for a.e. x ∈ ℝ that

(x) = Ψ(x, !)(Ψ(d, !)− eikdIn) = (x+ d)− eikd (x)

= ei(k+d)x

m+1∑j=0

(x+ d)juj(x+ d)− eikdeikxm+1∑j=0

xjuj(x)

= eikxm+1∑j=0

((x+ d)j − xj

)eikduj(x) = eikx

m+1∑j=1

j−1∑l=0

(j

l

)dj−lxleikduj(x)

= eikxm∑l=0

xl

(m∑j=l

(j

l

)dj+1−leikduj+1(x)

)= eikx

m∑l=0

xluj(x),

129

where uj =∑m

j=l

(jl

)dj+1−leikduj+1 ∈ (W 1,1

loc (ℝ))n ∩ (L1(T))n for j = 0, . . . ,m and um =

deikdum+1 ∕= 0. But this proves is a Floquet solution of the canonical ODEs in (4.13) with

wavenumber-frequency pair (k, !), Floquet multiplier � = eikd, and order m. Hence by the

induction hypothesis and the uniqueness portion of Proposition 33.(i) we conclude that is

a generalized eigenvector of Ψ(d, !) of order m + 1 corresponding to the eigenvalue �. But

since = (Ψ(d, !) − �In) this implies is a generalized eigenvector of Ψ(d, !) of order

m+ 2 corresponding to the eigenvalue �. This proves the statement for m+ 1. Therefore by

induction the statement is true for all m ∈ ℕ ∪ {0}.

To complete the proof of this theorem we must show that if is a generalized eigenvector

of Ψ(d, !) of order m + 1 corresponding to the eigenvalue � then for any k ∈ ℂ such

that � = eikd, = Ψ(⋅, !) is a Floquet solution of the canonical ODEs in (4.13) with

wavenumber-frequency pair (k, !), Floquet multiplier � = eikd, and order m.

Let be a generalized eigenvector of Ψ(d, !) of order m+ 1 corresponding to the eigenvalue

� where m ∈ ℕ∪{0}. Define := Ψ(⋅, !) . By Proposition 33.(i), is a nontrivial solution

of the canonical ODEs in (4.13) at the frequency ! ∈ Ω. We will now show that is a

Floquet solution with the desired properties.

To simplify notation we will denote the matricant by Φ := Ψ(⋅, !). Then by the Floquet-

Lyapunov Theorem there exists a matrix K ∈Mn(ℂ) and a function F ∈Mn(W 1,1loc (ℝ)) such

that for every x ∈ ℝ,

Φ(x) = F (x)eixK ,

where F (x + d) = F (x), F (0) = In, F−1(x) := F (x)−1 exist, and F−1 ∈ Mn(W 1,1loc (ℝ)).

Moreover, by the proof of the Floquet-Lyapunov Theorem we can assume that

K =1

idlog(Φ(d))

130

where, letting �1, . . . , �s denote the distinct eigenvalues of the monodromy matrix Φ(d),

log(z) is a branch of the logarithm which is analytic at each �l, for l = 1, . . . , s. Fix an

x ∈ ℝ. We define the function

f(z) := exdlog(z)

with the same domain as the function log(z). It follows that f is analytic at eigenvalues of

the monodromy matrix Φ(d) as well. By the Riesz-Dunford functional calculus as described

for matrices in [35, pp. 304–334, §9] we can define the matrix f(Φ(d)). But f(z) = eix(1idlog(z))

and so it follows by [35, pp. 324, §9.7, Theorem 2] that

f(Φ(d)) = eixK .

For each l = 1, . . . , s, let Pl ∈Mn(ℂ) denote the projection onto the generalized eigenspace of

the monodromy matrix Φ(d) corresponding to the eigenvalue �l. Without loss of generality

we may assume �1 = �. Let ml denote the index of the eigenvalue �l for l = 1, . . . , s. Let

fl,j denote the value of the jth derivative of f(z) at the eigenvalue �l for l = 1, . . . , s and

j = 0, . . . ,mk − 1. By spectral resolution of f(Φ(d)) (see [35, p. 314, §9.5, Theorem 1], [35,

p. 319, §9.5, Theorem 3], and [35, p. 321, §9.6, Theorem 1]) we have

f(Φ(d)) =s∑l=1

ml−1∑j=0

fl,jj!

(Φ(d)− �lIn)jPl

Now since by hypothesis is a generalized eigenvector of Φ(d) of order m+ 1 corresponding

to the eigenvalue �1 this implies

P1 = , Pl = 0, for l ∕= 1,

(Φ(d)− �1In)m ∕= 0, (Φ(d)− �1In)j = 0, for j > m.

131

Thus we have

f(Φ(d)) =m∑j=0

f1,j

j!(Φ(d)− �1In)j , f1,j =

djf

dzj

∣∣∣z=�1

.

But we have

djf

dzj

∣∣∣z=�1

=(xd

)(xd− 1)⋅ ⋅ ⋅(xd− (j − 1)

)e(

xd−(j−1)) log(�1), j = 1, . . . ,m.

We define the polynomials in the variable x by

p0(x) := 1, pj(x) :=(xd

)(xd− 1)⋅ ⋅ ⋅(xd− (j − 1)

)e−(j−1) log(�1).

Choose any k ∈ ℂ such that

�1 = eikd.

Then, since �1 = elog(�1) which follows from the fact log(z) is a branch of the logarithm

analytic at �1, there exist q ∈ ℤ such that

ikd = log(�1) + i2�q.

From these facts it follows that

eixK = f(Φ(d)) =m∑j=0

f1,j

j!(Φ(d)− �1In)j =

m∑j=0

exd

log(�1)pj(x)

j!(Φ(d)− �1In)j

= eikxm∑j=0

pj(x)

j!e−ix2�q/d(Φ(d)− �1In)j .

132

And from this we conclude

(x) = Φ(x) = F (x)eixK = eikxm∑j=0

pj(x)

j!e−ix2�q/dF (x)(Φ(d)− �In)j , (4.63)

for every x ∈ ℝ. But vj(⋅) := 1j!e−i(⋅)2�q/dF (⋅)(Φ(d)−�In)j ∈ (W 1,1

loc (ℝ))n∩ (L1(T))n and the

polynomial pj(⋅) has degree j, for j = 0, . . . ,m. Moreover, vm(0) = 1m!

(Φ(d) − �In)m ∕= 0

and � = eikd. From these facts and the representation (4.63) we conclude that for any k ∈ ℂ

such that � = eikd, is Floquet solution of the canonical ODEs in (4.13) with wavenumber-

frequency pair (k, !), Floquet multiplier � = eikd, and order m. This completes the proof of

Proposition 35.

Proof. [Corollary 36] Fix an ! ∈ Ω. Let { j}nj=1 be a Jordan basis for the matrix Ψ(d, !).

Then for j = 1, . . . , n, we can define lj, �j, and kj to be such that j is a generalized

eigenvalue of Ψ(d, !) of order lj corresponding to the eigenvalue �j = eikjd. We define

{ j}nj=1 by j := Ψ(⋅, !) j for j = 1, . . . , n. Then since { j}nj=1 is a basis for (ℂ)n it follows

by 33.(i) and 33.(v) that the set of solutions of the canonical ODEs in (4.13) at the frequency

! is a vector space over ℂ and { j}nj=1 is a basis for that space. Moreover, by Theorem 35,

j is a Floquet solution of the canonical ODEs in (4.13)with wavenumber-frequency pair

(kj, !), Floquet multiplier �j = eikjd, and order lj − 1, for j = 1, . . . , n. This completes the

proof.

Proof. [Corollary 37 & Corollary 38] Let ℬ denote the Bloch variety of the canonical

ODEs in (4.13), i.e.,

ℬ := {(k, !) ∈ ℂ× Ω ∣ (k, !) is the wavenumber-frequency pair of some nontrivial

Bloch solution of the canonical ODEs in (4.13)}.

133

Define the function D : ℂ× Ω→ ℂ by

D(k, !) := det(eikdIn −Ψ(d, !)

)and the set C ⊆ ℂ× Ω by

C := {(k, !) ∈ ℂ× Ω ∣ D(k, !) = 0}.

Now Corollary 37 follows from Corollary 38 since D(k+ 2�/d, !) = D(k, !). Hence to prove

these corollaries we need only show the function D is a nonconstant holomorphic function

and ℬ = C.

We will now prove ℬ = C. Let (k, !) ∈ ℬ. Then there exists a ∕= 0 which is a Bloch

solution of the canonical ODEs in (4.13) with wavenumber-frequency pair (k, !). But this

implies by Definition 19 that is a Floquet solution of the canonical ODEs in (4.13) with

wavenumber-frequency pair (k, !), Floquet multiplier � = eikd, and order 1. By Theorem 35

this implies = Ψ(⋅, !) where is an eigenvector of the monodromy matrix Ψ(⋅, !) with

� = eikd as its corresponding eigenvalue. Thus we must have det(eikdIn −Ψ(d, !)

)= 0 and

hence (k, !) ∈ C. This proves ℬ ⊆ C.

To prove the reverse inclusion, let (k, !) ∈ C. Then det(eikdIn −Ψ(d, !)

)= 0 and this

implies � = eikd is an eigenvalue of the monodromy matrix Ψ(d, !). Let be an eigenvector of

the monodromy matrix Ψ(d, !) corresponding to the eigenvalue � = eikd. Then Theorem 35

implies := Ψ(⋅, !) is a Floquet solution of the canonical ODEs in (4.13) with wavenumber-

frequency pair (k, !), Floquet multiplier � = eikd, and order 1. This and Lemma 57 imply

is a nontrivial Bloch solution of the canonical ODEs in (4.13) with wavenumber-frequency

pair (k, !) and hence (k, !) ∈ ℬ. This proves C ⊆ ℬ. Therefore ℬ = C.

134

We will now prove the function D : ℂ× Ω→ ℂ is a nonconstant function. First, f(�, !) :=

det (�In −Ψ(d, !)) is a monic polynomial of degree n in the variable � for each ! ∈ ℂ. Thus

f : ℂ×Ω→ ℂ is a nonconstant function. Now D(k, !) = f(eikd, !) and this implies D must

be a nonconstant function for otherwise we could fix an !0 ∈ Ω and then there would exist

c ∈ ℂ such that f(eikd, !0) − c = 0 for every k ∈ ℂ implying, since f(�, !0) − c is a monic

polynomial of degree n in the variable �, the function g(k) := eikd maps into a finite set of

values in ℂ which is a contradiction. Thus we have proven the function D is a nonconstant

function.

We complete the proof of these corollaries now by proving the function D : ℂ × Ω → ℂ

is a holomorphic function. By Lemma 80 we have O(Ω,Mn(ℂ)) = Mn(O(Ω,ℂ)). Thus

by Proposition 33.(iii) we have Ψ(d, ⋅) ∈ Mn(O(Ω,ℂ)), i.e., the entries of the monodromy

matrix function Ψ(d, ⋅) are holomorphic functions of frequency. This implies the entries of

the matrix M(k, !) := eikdIn − Ψ(d, !) are holomorphic functions of the variable (k, !) in

the domain ℂ×Ω. And since the function F (B) := det(B) is a polynomial in the entries of

the matrix B ∈Mn(ℂ) this implies D(k, !) = F (M(k, !)) is a holomorphic functions of the

variable (k, !) in the domain ℂ× Ω. This completes the proof.

Proofs for Section 4.2.2

We begin with proving a key result in the study of the energy flux for canonical ODEs.

Proof. [Proposition 39] Fix an ! ∈ Ωℝ. Then by Proposition 33 we have Ψ(⋅, !) ∈

Mn(W 1,1loc (ℝ)), by Lemma 75 we have Ψ(⋅, !)∗ ∈ Mn(W 1,1

loc (ℝ)), and hence by Lemma 73 we

have Ψ(⋅, !)∗iJ Ψ(⋅, !) ∈Mn(W 1,1loc (ℝ)). For simplification of notation let Φ := Ψ(⋅, !).

To complete the proof we must show Φ∗iJ Φ = iJ in Mn(W 1,1loc (ℝ)). First, it follows from

Proposition 33 that Φ′ = J −1A(⋅, !)Φ in Mn(L1loc(ℝ)). Second, by our assumptions we

135

have (J −1)∗ = −J −1 and A(⋅, !)∗ = A(⋅, !) in Mn(L1loc(ℝ)). Thus by these facts and by

Lemmas 75 and 73 we have

(Φ∗iJ Φ)′ = (Φ′)∗iJ Φ + Φ∗iJ Φ′ = (J −1A(⋅, !)Φ)∗iJ Φ + Φ∗iA(⋅, !)Φ

= −iΦ∗A(⋅, !)Φ + iΦ∗A(⋅, !)Φ = 0.

It follows from this that there exists a constant matrix C ∈Mn(ℂ) such that

Φ∗iJ Φ = C.

But since Φ(0) = Ψ(0, !) = In this implies C = iJ and hence

Φ∗iJ Φ = iJ .

This completes the proof.

Proof. [Lemma 41] Let !0 ∈ Ωℝ. By Lemma 76 and the hypotheses the Hamilto-

nian satisfies A(⋅, !)∗ = A(⋅, !) as an element of Mn(L1(T))) for every ! ∈ Ωℝ and A ∈

O(Ω,Mn(L1(T))) as a function of frequency. By definition since A! is derivative of the A

with respect to frequency in the Banach space Mn(L1(T)) we have A!(⋅, !) ∈Mn(L1(T)) for

every ! ∈ Ω and

lim!→!0

∥(! − !0)−1(A(⋅, !)− A(⋅, !0))− A!(⋅, !0)∥L1(T) = 0.

It then follows from these facts and Lemma 76 that

0 = lim!→!0

!∈Ωℝ

∥(! − !0)−1(A(⋅, !)− A(⋅, !0))− A!(⋅, !0)∥L1(T)

= lim!→!0

!∈Ωℝ

∥((! − !0)−1(A(⋅, !)− A(⋅, !0))− A!(⋅, !0)

)∗ ∥L1(T)

136

= lim!→!0

!∈Ωℝ

∥(! − !0)−1(A(⋅, !)− A(⋅, !0))− A!(⋅, !0)∗∥L1(T).

From which it follows that A!(⋅, !0)∗ = A!(⋅, !0) in Mn(L1(T))). This completes the proof.

Proof. [Theorem 42] Let !0 ∈ Ωℝ. Then Proposition 33 implies Ψ(⋅, !0), Ψ!(⋅, !0) ∈

Mn(W 1,1loc (ℝ)) and by Lemma 75 have Ψ(⋅, !0)∗ ∈ Mn(W 1,1

loc (ℝ)). Thus by Lemma 73 we

have Ψ(⋅, !0)∗J Ψ!(⋅, !0) ∈ Mn(W 1,1loc (ℝ)). Thus by notation 4.5.(xxix) we must have

(Ψ(⋅, !0)∗J Ψ!(⋅, !0))′ ∈ Mn(L1loc(ℝ)). Now since A!(⋅, !0) ∈ Mn(L1(T))) ⊆ Mn(L1

loc(ℝ))

then it follows from Lemma 84 that we also have Ψ(⋅, !0)∗A!(⋅, !0)Ψ(⋅, !0) ∈Mn(L1loc(ℝ)).

Our next goal is to show that as elements of Mn(L1loc(ℝ)),

Ψ(⋅, !0)∗A!(⋅, !0)Ψ(⋅, !0) = (Ψ(⋅, !0)∗J Ψ!(⋅, !0))′.

We begin by defining two operators that will play a key role in our proof. The first is the

integral map defined by

(ℐB)(x) :=

∫ x

0

B(t)dt, B ∈ (L1loc(ℝ))n, x ∈ ℝ.

By Lemma 87, the map ℐ : (L1loc(ℝ))n → (W 1,1

loc (ℝ))n is well-defined and

(ℐf)′ = f,

for every f ∈ (W 1,1loc (ℝ))n. The second map is defined for each ! ∈ Ω by

(S(!)f)(x) := Ψ(x, !)

∫ x

0

Ψ(t, !)−1f(t)dt, f ∈ (L1loc(ℝ))n, x ∈ ℝ.

137

Lemma 58 For each ! ∈ Ω, the map S(!) : (L1loc(ℝ))n → (W 1,1

loc (ℝ))n is a right inverse of

the map T (!) : (W 1,1loc (ℝ))n → (L1

loc(ℝ))n defined in (4.47), i.e., for every f ∈ (L1loc(ℝ))n we

have

T (!)S(!)f = f.

Proof. Let ! ∈ Ω. Lets first prove the map S(!) : (L1loc(ℝ))n → (W 1,1

loc (ℝ))n is well-

defined. By Proposition 33.(v) it follows that Ψ−1(x, !) := Ψ(x, !)−1 exists for every

x ∈ ℝ and Ψ−1(⋅, !) ∈ Mn(W 1,1loc (ℝ)). Let f ∈ (L1

loc(ℝ))n. Then it follows by Lemma

84 that Ψ−1(⋅, !)f ∈ (L1loc(ℝ))n and applying the integral map to this yields ℐ(Ψ−1(⋅, !)f) ∈

(W 1,1loc (ℝ))n. By Lemma 73 and the definition of S(!) we find that

S(!)f = Ψ(⋅, !)ℐ(Ψ−1(⋅, !)f) ∈ (W 1,1loc (ℝ))n.

Thus the map is well-defined.

Next, recall that the map T (!) : (W 1,1loc (ℝ))n → (L1

loc(ℝ))n in (4.47) is

T (!) := ′ −J −1A(⋅, !) , ∈ (W 1,1loc (ℝ))n.

We now show S(!) is a right inverse of T (!). Let f ∈ (L1loc(ℝ))n then by Lemma 73 we have

(S(!)f)′ = (Ψ(⋅, !)ℐ(Ψ−1(⋅, !)f))′ = Ψ(⋅, !)′ℐ(Ψ−1(⋅, !)f) + Ψ(⋅, !)(ℐ(Ψ−1(⋅, !)f))′

= J −1A(⋅, !)Ψ(⋅, !)ℐ(Ψ−1(⋅, !)f) + Ψ(⋅, !)Ψ−1(⋅, !)f = J −1A(⋅, !)S(!)f + f.

Hence it follows that

T (!)S(!)f = (S(!)f)′ −J −1A(⋅, !)S(!)f = f.

138

Thus S(!) is a right inverse of T (!). This completes the proof.

Let ∈ ℂn. Define � := Ψ!(⋅, !0) and f := J −1A!(⋅, !0)Ψ(⋅, !0) . We will now show

that � = S(!0)f . It follows from Lemma 84 that f ∈ (L1loc(ℝ))n. It follows by Proposition

33.(iv) that � ∈ (W 1,1loc (ℝ))n and satisfies

� = Ψ!(⋅, !0) = ℐ(J −1A!(⋅, !0)Ψ(⋅, !0) + J −1A(⋅, !0)Ψ!(⋅, !0)

)= ℐ

(f + J −1A(⋅, !0)�

)since J −1A!(⋅, !0)Ψ(⋅, !0) ∈ Mn(L1

loc(ℝ)) by Lemma 84. Hence by Lemma 87 it follows

that

T (!0)� = �′ −J −1A(⋅, !0)� = f.

But by Lemma 58 we also have T (!0)S(!0)f = f with S(!0)f ∈ (W 1,1loc (ℝ))n. Hence

:= �− S(!0)f ∈ (W 1,1loc (ℝ))n and satisfies

T (!0) = 0,

that is, ∈ ker(T (!0)). Thus by Corollary 54 and Proposition 33.(i) there exists a unique

∈ ℂn such that = Ψ(⋅, !0) . In fact, = (0) = �(0)−(S(!0)f)(0). But (S(!)f)(0) = 0

and �(0) = Ψ!(0, !0) = 0. Thus we have = 0 and so � = S(!0)f as desired.

Now it follows from what we have just shown that for every ∈ ℂn,

Ψ!(⋅, !0) = S(!0)(J −1A!(⋅, !0)Ψ(⋅, !0) )

139

from which it follows by the definition of S(!0) and Lemma 87 that

Ψ!(x, !0) = Ψ(x, !0)

∫ x

0

Ψ(t, !0)−1J −1A!(t, !0)Ψ(t, !0) dt

= Ψ(x, !0)

∫ x

0

Ψ(t, !0)−1J −1A!(t, !0)Ψ(t, !0)dt , ∀x ∈ ℝ.

But since this is true for every ∈ ℂn this implies

Ψ!(x, !0) = Ψ(x, !0)

∫ x

0

Ψ(t, !0)−1J −1A!(t, !0)Ψ(t, !0)dt, ∀x ∈ ℝ.

Now if we multiply both sides by Ψ(x, !)∗J and apply Proposition 39 we get

Ψ(x, !0)∗J Ψ!(x, !0) = Ψ(x, !)∗J Ψ(x, !0)

∫ x

0

Ψ(t, !0)−1J −1A!(t, !0)Ψ(t, !0)dt

= J

∫ x

0

Ψ(t, !0)−1J −1A!(t, !0)Ψ(t, !0)dt =

∫ x

0

Ψ(t, !0)∗A!(t, !0)Ψ(t, !0)dt

for every x ∈ ℝ. It follows from this and Lemma 87 that

(Ψ(⋅, !0)∗J Ψ!(⋅, !0))′ = Ψ(⋅, !0)∗A!(⋅, !0)Ψ(⋅, !0)

and

1

d

∫ d

0

Ψ(t, !0)∗A!(t, !0)Ψ(t, !0)dt =1

dΨ(d, !)∗J Ψ!(d, !0).

This completes the proof.

Proof. [Corollary 43] Let ! ∈ Ωℝ and 1, 2 ∈ ℂn. Then by Theorem 42 we have

Ψ(⋅, !)∗J Ψ!(⋅, !) 1 ∈ (W 1,1loc (ℝ))n

Ψ(⋅, !)∗A!(⋅, !)Ψ(⋅, !) 1 ∈ (L1loc(ℝ))n

(Ψ(⋅, !)∗J Ψ!(⋅, !) 1)′ = Ψ(⋅, !)∗A!(⋅, !)Ψ(⋅, !) 1

140

Hence it follows from this and Lemma 75 that ⟨Ψ(⋅, !)∗J Ψ!(⋅, !) 1, 2⟩ℂ ∈ W 1,1loc (ℝ),

⟨A!(⋅, !) 1, 2⟩ℂ ∈ L1loc(ℝ), and ⟨Ψ(⋅, !)∗J Ψ!(⋅, !) 1, 2⟩ℂ

′ = ⟨A!(⋅, !) 1, 2⟩ℂ. And by

Theorem 42 we conclude that

1

d

∫ d

0

⟨A!(x, !) 1(x), 2(x)⟩ℂdx =1

d⟨Ψ(d, !)∗J Ψ!(d, !) 1, 2⟩ℂ.

This completes the proof.

Proofs for Section 4.2.3

We begin by proving the sesquilinear form defined in (4.31) is a Hermitian form.

Proof. [Lemma 44] Let (k, !) ∈ ℬℝ. By Lemma 41, for any 1, 2 ∈ ker (eikdIn −Ψ(d, !)),

if we let 1 := Ψ(⋅, !) 1, 2 := Ψ(⋅, !) 2 then we have

q(k,!)

( 1, 2) =1

d

∫ d

0

⟨A!(x, !) 1(x), 2(x)⟩ℂdx =1

d

∫ d

0

⟨A!(x, !) 1(x), 2(x)⟩ℂdx

=1

d

∫ d

0

⟨ 2(x), A!(x, !) 1(x)⟩ℂdx =1

d

∫ d

0

⟨A!(x, !) 2(x), 1(x)⟩ℂdx = q(k,!)

( 2, 1),

implying q(k,!)

( 1, 1) ∈ ℝ. This completes the proof.

Proof. [Proposition 45] By Definition 23, the canonical ODEs in (4.13) are of definite

type at a point (k0, !0) ∈ ℬℝ if and only if

1

d

∫ d

0

⟨A!(x, !0) (x), (x)⟩ℂdx ∕= 0

for any nontrivial Bloch solution of the canonical ODEs in (4.13) with wavenumber-

frequency (k0, !0). But by Theorem 35, is a nontrivial Bloch solution of the canonical

ODEs in (4.13) with wavenumber-frequency (k0, !0) if and only if = Ψ(⋅, !0) where

141

is an eigenvector of the monodromy matrix Ψ(d, !0) corresponding to the eigenvalue eik0d.

And hence the canonical ODEs in (4.13) are of definite type at a point (k0, !0) ∈ ℬℝ if and

only if ker (eik0dIn −Ψ(d, !0)) ∕= {0} and

q(k0,!0)

( , ) ∕= 0, for every nonzero ∈ ker (eik0dIn −Ψ(d, !0)). (4.64)

Thus to complete the proof we need only show that (4.64) implies q(k0,!0)

, as defined in (4.31),

is a definite sesquilinear form which is bounded.

Suppose that (4.64) is true. Then it follows from Lemma 44 that q(k0,!0)

( , ) ∈ ℝ/{0}

for every nonzero ∈ ker (eik0dIn −Ψ(d, !0)). Suppose there existed nonzero −, + ∈

ker (eik0dIn −Ψ(d, !0)) such that q(k0,!0)

( − −) < 0 < q(k0,!0)

( +, +). Then we can define a

continuous real-valued function f : [0, 1]→ ℝ by

f(x) := q(k0,!0)

(x + + (1− x) −, x + + (1− x) −).

Now f(0) < 0 < f(1) and so by the intermediate value theorem that there exists x0 ∈

(0, 1) such that f(x0) = 0 implying by hypothesis that x0 + + (1 − x0) − = 0. But this

implies 0 < x20q(k0,!0)( +, +) = (1 − x0)2q

(k0,!0)( −, −) < 0, a contradiction. Thus we

conclude that sgn(q(k0,!0)

( , )) =: sgn(q(k0,!0)

) is independent of the choice of nonzero in

ker (eik0dIn −Ψ(d, !0)).

We define a sesquilinear form by

⟨ 1, 2⟩(k0,!0) := sgn(q(k0,!0)

)q(k0,!0)

( 1, 2), 1, 2 ∈ ker (eik0dIn −Ψ(d, !0)).

It follows from what we have just shown that ⟨ , ⟩(k0,!0) is an inner product on the finite-

142

dimensional vector space ker (eik0dIn −Ψ(d, !0)) and so

∣∣ ∣∣(k0,!0) := ⟨ , ⟩12

(k0,!0) = ∣q(k0,!0)

( , )∣12 , ∈ ker (eik0dIn −Ψ(d, !0)),

defines a norm on this space as does the Euclidean norm ∣∣ ⋅ ∣∣ℂ. But since any two norms

on a finite dimensional vector space are equivalent this implies there exists C1, C2 > 0 such

that

C1∣∣ ∣∣2ℂ ≤ ∣∣ ∣∣2(k0,!0) ≤ C2∣∣ ∣∣2ℂ

for every ∈ ker (eik0dIn −Ψ(d, !0)). This completes the proof.

Proof. [Lemma 46] Let {(kj, !j)}j∈ℕ be a sequence in ℬℝ and { j}j∈ℕ a sequence in ℂn

such that j ∈ ker(eikjdIn −Ψ(d, !j)), ∣∣ j∣∣ℂ = 1 for all j ∈ ℕ and

(kj, !j)∣∣⋅∣∣ℂ−−→ (k0, !0) and j

∣∣⋅∣∣ℂ−−→ 0, as j →∞,

for some k0, !0 ∈ ℂ and 0 ∈ ℂn. We will now show that ∣∣ 0∣∣ℂ = 1, 0 ∈ ker(eik0dIn −

Ψ(d, !0)), and (k0, !0) ∈ ℬℝ.

First, since j∣∣⋅∣∣ℂ−−→ 0 as j →∞, then

1 = limj→∞∣∣ j∣∣ℂ = ∣∣ 0∣∣ℂ.

Second, if we define a matrix-valued function by M(k, !) := eikdIn−Ψ(d, !), (k, !) ∈ ℂ×Ω

then this function is continuous since

∣∣M(k, !)−M(k0, !0)∣∣ℂ ≤ ∣∣M(k, !)−M(k0, !)∣∣ℂ + ∣∣M(k0, !)−M(k0, !0)∣∣ℂ

= ∣eikd − eik0d∣n12 + ∣∣Ψ(d, !)−Ψ(d, !0)∣∣ℂ

143

and by Proposition 33.(iii) we have Ψ(d, ⋅) ∈ O(Ω,Mn(ℂ) implying ∣∣Ψ(d, !)−Ψ(d, !0)∣∣ℂ →

0 as ! → !0. Thus we conclude from this that

∣∣M(k0, !0) 0∣∣ℂ = limj→∞∣∣M(kj, !j) j −M(k0, !0) 0∣∣ℂ

≤ limj→∞∣∣M(kj, !j) j −M(k0, !0) j∣∣ℂ + lim

j→∞∣∣M(k0, !0) j −M(k0, !0) 0∣∣ℂ

≤ limj→∞∣∣M(kj, !j)−M(k0, !0)∣∣ℂ∣∣ j∣∣ℂ + lim

j→∞∣∣M(k0, !0)∣∣ℂ∣∣ j − 0∣∣ℂ

= 0

and hence 0 ∈ ker(eik0dIn − Ψ(d, !0)). It now follows from this and Corollary 38 that

(k0, !0) ∈ ℬℝ.

Now we complete the proof by showing

q(kj,!j)

( j, j)∣⋅∣−→ q

(k0,!0)( 0, 0), as j →∞. (4.65)

First, it follows from Corollary 43 that for any (k, !) ∈ ℬℝ and for any �1, �2 ∈ ker(eikdIn−

Ψ(d, !)) we have

q(k,!)

(�1, �2) =1

d⟨Ψ(d, !)∗J Ψ!(d, !)�1, �2⟩ℂ.

Thus we extend the definition of q(k,!)

to all of ℂn × ℂn in the obvious way by

q(k,!)

(�1, �2) :=1

d⟨Ψ(d, !)∗J Ψ!(d, !)�1, �2⟩ℂ, �1, �2 ∈ ℂn.

It follows from this definition that q(k,!)

: ℂn × ℂn → ℂ is a continuous sesquilinear form

since

∣q(k,!)

(�1, �2)∣ ≤ d−1∣∣Ψ(d, !)∗J Ψ!(d, !)∣∣ℂ∣∣�1∣∣ℂ∣∣�2∣∣ℂ, ∀�1, �2 ∈ ℂn.

144

Hence for any (k, !) ∈ ℬℝ and any � ∈ ℂn we have the inequality

∣q(k,!)

(�, �)− q(k0,!0)

( 0, 0)∣ ≤ ∣q(k,!)

(�, �)− q(k,!)

(�, 0)∣

+ ∣q(k,!)

(�, 0)− q(k0,!0)

(�, 0)∣

+ ∣q(k0,!0)

(�, 0)− q(k0,!0)

( 0, 0)∣

and the inequalities

∣q(k,!)

(�, �)− q(k,!)

(�, 0)∣ = ∣q(k,!)

(�, � − 0)∣

≤ d−1∣∣Ψ(d, !)∗J Ψ!(d, !)∣∣ℂ∣∣�∣∣ℂ∣∣� − 0∣∣ℂ,

∣q(k0,!0)

(�, 0)− q(k0,!0)

( 0, 0)∣ = ∣q(k0,!0)

(� − 0, 0)∣

≤ d−1∣∣Ψ(d, !)∗J Ψ!(d, !)∣∣ℂ∣∣� − 0∣∣ℂ∣∣ 0∣∣ℂ,

∣q(k,!)

(�, 0)− q(k0,!0)

(�, 0)∣ = d−1∣∣Ψ(d, !)∗J Ψ!(d, !)

−Ψ(d, !0)∗J Ψ!(d, !0)∣∣ℂ∣∣�∣∣ℂ∣∣ 0∣∣ℂ

And thus it follows from these inequalities that (4.65) is true if

Ψ(d, !)∗J Ψ!(d, !)∣∣⋅∣∣ℂ−−→ Ψ(d, !0)∗J Ψ!(d, !0), as !

∣⋅∣−→ !0.

But by Lemma 77, multiplication of matrices in Mn(ℂ) is a continuous bilinear map and so

in order to prove the latter statement we need only show

Ψ(d, !)∗∣∣⋅∣∣ℂ−−→ Ψ(d, !0)∗ and Ψ!(d, !)

∣∣⋅∣∣ℂ−−→ Ψ!(d, !0), as !∣⋅∣−→ !0.

145

Convergence of the first limit follows from the fact that the adjoint map ()∗ : Mn(ℂ) →

Mn(ℂ) is a norm preserving map, i.e., ∣∣B∗∣∣ℂ = ∣∣B∣∣ℂ for everyB ∈Mn(ℂ), since ∣∣Ψ(d, !)∗−

Ψ(d, !0)∗∣∣ℂ = ∣∣(Ψ(d, !)−Ψ(d, !0))∗∣∣ℂ = ∣∣Ψ(d, !)−Ψ(d, !0)∣∣ℂ → 0 as ! → !0. Now for the

other limit, by Lemma 81 since Ψ(d, ⋅) ∈ O(Ω,Mn(ℂ)) and, by Proposition 33.(iii), Ψ!(d, ⋅)

is its derivative in the ∣∣ ⋅ ∣∣ℂ norm then Ψ!(d, ⋅) ∈ O(Ω,Mn(ℂ)). But this implies Ψ!(d, ⋅) is

a continuous matrix-valued function on Ω in the ∣∣ ⋅ ∣∣ℂ norm and so Ψ!(d, !)∣∣⋅∣∣ℂ−−→ Ψ!(d, !0)

as !∣⋅∣−→ !0. Therefore (4.65) is true and this completes the proof.

Proof. [Theorem 47] Suppose (k0, !0) ∈ ℬℝ is a point of definite type for the canonical

ODEs in (4.13). We will first show that there exists an r > 0 such that every (k, !) ∈

B((k0, !0), r)∩ℬℝ is a point of definite type for the canonical ODEs in (4.13). We will prove

this by contradiction. Suppose the statement was not true. Then we can find a sequence of

points {(kj, !j)}∞j=1 ⊆ ℬℝ and a sequence { j}∞j=1 ⊆ (W 1,1loc (ℝ))n such that j is a nontrivial

Bloch solution of the canonical ODEs in (4.13) with wavenumber-frequency pair (kj, !j) and

1

d

∫ d

0

⟨A!(x, !) j(x), j(x)⟩ℂdx = 0,

for every j ∈ ℕ. By Theorem 35 there exists a nonzero j ∈ ker(eikjdIn−Ψ(d, !j)) such that

j = Ψ(d, !j) j, for every j ∈ ℕ. Let �j := j∣∣ j ∣∣ℂ

, for j ∈ ℕ. Then {�j}∞j=1 is a sequence

of vectors with unit norm in ℂn and hence since {� ∈ ℂn : ∣∣�∣∣ℂ = 1} is a compact set

this implies there exists a convergent subsequence {�jl}∞l=1 such that �jl

∣∣⋅∣∣ℂ−−→ �0 as l → ∞

for some �0 ∈ ℂn with ∣∣�0∣∣ℂ = 1. Thus since {(kjl , !jl)}l∈ℕ ⊆ ℬℝ, {�jl}j∈ℕ ⊆ ℂn such that

�jl ∈ ker(eikjldIn −Ψ(d, !jl)) with ∣∣�jl ∣∣ℂ = 1 for all l ∈ ℕ and

(kjl , !jl)∣∣⋅∣∣ℂ−−→ (k0, !0) and �jl

∣∣⋅∣∣ℂ−−→ �0, as l→∞

146

then by Lemma 46, �0 ∈ ker(eik0dIn −Ψ(d, !0)) with ∣∣�0∣∣ℂ = 1 and

q(kjl

,!jl)(�jl , �jl)

∣⋅∣−→ q(k0,!0)

(�0, �0) as l→∞.

But for every l ∈ ℕ we have

q(kjl

,!jl)(�jl , �jl) = ∣∣ jl ∣∣

−2ℂ q

(kjl,!jl

)( jl , jl)

= ∣∣ jl ∣∣−2ℂ

1

d

∫ d

0

⟨A!(x, !) jl(x), jl(x)⟩ℂdx = 0

implying

q(k0,!0)

(�0, �0) = 0, ∣∣�0∣∣ℂ = 1, �0 ∈ ker(eik0dIn −Ψ(d, !0)).

By Proposition 45 this contradicts that (k0, !0) ∈ ℬℝ is a point of definite type for the

canonical ODEs in (4.13). Thus we conclude that there exists an r > 0 such that every

(k, !) ∈ B((k0, !0), r) ∩ ℬℝ is a point of definite type for the canonical ODEs in (4.13).

Now let {(kj, !j)}∞j=1 ⊆ ℬℝ be any sequence converging to (k0, !0) in the Euclidean norm

∣∣ ⋅ ∣∣ℂ. Then since the set {� ∈ ℂn : ∣∣�∣∣ℂ = 1} is compact we can find a sequence { j}∞j=1

such that j ∈ ker(eikjdIn −Ψ(d, !j)) with ∣∣ j∣∣ℂ = 1 for all j ∈ ℕ and

j∣∣⋅∣∣ℂ−−→ 0, as j →∞

for some 0 ∈ {� ∈ ℂn : ∣∣�∣∣ℂ = 1}. From what we just proved there exists a J ∈ ℕ such

that if j ≥ J then (kj, !j) is a point of definite type for the canonical ODEs in (4.13). Thus

by Proposition 45 and Lemma 46 it follows that 0 ∈ ker(eik0dIn−Ψ(d, !0)) with ∣∣ 0∣∣ℂ = 1

and

sgn(q(kj,!j)

) =q(kj,!j)

( j, j)

∣q(kj,!j)

( j, j)∣∣⋅∣−→

q(k0,!0)

( 0, 0)

∣q(k0,!0)

( 0, 0)∣= sgn(q

(k0,!0)), as j →∞.

147

From which we conclude that for j sufficiently large we must have sgn(q(kj,!j)

) = sgn(q(k0,!0)

).

Therefore, since this is true for an arbitrary sequence in ℬℝ, we conclude that there exists an

r > 0 such that every (k, !) ∈ B((k0, !0), r)∩ℬℝ is a point of definite type for the canonical

ODEs in (4.13) and sgn(q(k,!)

) = sgn(q(k0,!0)

). This completes the proof.

Proofs for Section 4.2.4

We now begin proving our main results on the perturbation theory for canonical ODEs. Our

proof relies heavily on the spectral perturbation theory for holomorphic matrix functions

and the reader may wish to review chapter 3 before preceding.

Proof. [Theorem 48] Let (k0, !0) ∈ ℬℝ be a point of definite type for the canonical ODEs

in (4.13). Let g be the number of Jordan blocks (geometric multiplicity) in the Jordan

form of the monodromy matrix Ψ(d, !0) corresponding to the eigenvalue �0 = eik0d and

m1 ≥ ⋅ ⋅ ⋅ ≥ mg ≥ 1 the dimensions of each of those Jordan blocks (partial multiplicities).

Define m := m1 + ⋅ ⋅ ⋅+mg (algebraic multiplicity). Let � > 0 be given.

Choose an r0 > 0 such that 0 < r0 ≤ � and B(!0, r0) ∈ Ω. Define the function L : U →

Mn(ℂ) by

L(!, k) := Ψ(d, !)− eikdIn. (4.66)

where U := B(!0, r0)× B(k0, r0). By definition of a holomorphic matrix-valued function of

two variables, see 3.2.(i), and the fact Ψ(d, ⋅) ∈ O(B(!0, r0),Mn(ℂ)), it follows that

L ∈ O(U,Mn(ℂ)). (4.67)

We will now use the results from section 3.2 and give a spectral perturbation analysis of the

148

holomorphic matrix function L(⋅, k) near the point (!0, k0) ∈ �(L) = ℬ−1.

First, since B(!0, r0)∗ = B(!0, r0), the adjoint of the holomorphic function Ψ(d, !), see

Definition 14, is

Ψ∗(d, !) = Ψ(d, !)∗, ! ∈ B(!0, r0), (4.68)

and by Lemma 24 we have

Ψ∗(d, ⋅) ∈ O(B(!0, r0),Mn(ℂ)). (4.69)

Next, since U∗ = U , the adjoint L∗ of the holomorphic function of two variables L, see

Definition 15, is

L∗(!, k) = L(!, k)∗, (!, k) ∈ U, (4.70)

and by Lemma 26 we have

L∗ ∈ O(U,Mn(ℂ)). (4.71)

We now prove series of lemmas which relate the spectrum of L to the spectrum of its adjoint

L∗.

Lemma 59 For every (!, k) ∈ U we have

L∗(!, k)J = −e−ikdΨ∗(d, !)JL(!, k). (4.72)

149

Proof. By Proposition 39 we have

Ψ∗(d, !)J Ψ(d, !) = Ψ(d, !)∗J Ψ(d, !) = J

for every B(!0, r0) ∩ ℝ ⊆ Ωℝ. Thus if (!, k) ∈ U ∩ ℝ2 then

− e−ikdΨ∗(d, !)JL(!, k) = −e−ikdΨ(d, !)∗J (Ψ(d, !)− eikdIn)

= Ψ(d, !)∗J − e−ikdJ = (Ψ(d, !)∗ − e−ikd)J

= L∗(!, k)J .

But for the matrix-valued function f(!, k) := −e−ikdΨ∗(d, !)JL(!, k)−L∗(!, k)J we have

f ∈ O(U,Mn(ℂ)) and its identically equal to zero on U∩ℝ2 where U = B(!0, r0)×B(k0, r0).

By Lemma 17, f is an analytic function of two variables identically equal to zero on U ∩ℝ2

hence using the power series expansion of f at the point (!0, k0) and the fact f = 0 on

U ∩ ℝ2 we can conclude f = 0 in an open connected set of ℂ2 in U which contains (!0, k0).

Thus, since U is an open connected set on which f is analytic, by analytic continuation we

conclude that f = 0 on U . This completes the proof of the lemma.

The following two lemmas are the key to the proof of this theorem.

Lemma 60 If !(k) is an eigenvalue Puiseux series of L(⋅, k) expanded about k0 with limit

point !0 then !(k) and !∗(k), the adjoint of the Puiseux series !(k), both are eigenvalue

Puiseux series of L∗(⋅, k) and L(⋅, k) expanded about k0 with limit point !0.

Proof. By Lemma 59, the fact J and Ψ(d, !) are invertible matrices for every ! ∈ Ω,

implies the holomorphic matrix function L(⋅, k) and its adjoint L∗(⋅, k) are equivalent, i.e,

L(⋅, k) ∼ L∗(⋅, k), at ! in the sense of holomorphic matrix functions from chapter 3 for every

(!, k) ∈ U . In particular, L(⋅, k) and its adjoint L∗(⋅, k) have the same eigenvalues for every

150

k ∈ B(k0, r0), i.e., for every k ∈ B(k0, r0)

�(L∗(⋅, k)) = �(L(⋅, k)).

This implies if !(k) is an eigenvalue Puiseux series of L(⋅, k) expanded about k0 with limit

point !0 then by Lemma 28 its adjoint !∗(k) is an eigenvalue Puiseux series of L∗(⋅, k)

expanded about k0 = k0 with limit point !0 = !0. But since �(L∗(⋅, k)) = �(L(⋅, k)) then

all the values of the branches of the Puiseux series !∗(k) and !(k) are eigenvalues of both

L∗(⋅, k) and L(⋅, k) which, by the definition of eigenvalue Puiseux series, means !∗(k) and

!(k) are also eigenvalue Puiseux series of L∗(⋅, k) expanded about k0 with limit point !0.

This completes the proof.

Lemma 61 If !(k) is an eigenvalue Puiseux series of L(⋅, k) expanded about k0 with limit

point !0 and � is generating eigenvector of L(⋅, k) at the point (!0, k0) and associated with

!(k) then J � is a generating eigenvector of L∗(⋅, k) at the point (!0, k0) and associated with

!(k). Moreover,

⟨L!(!0, k0)�,J �⟩ℂ ∕= 0. (4.73)

Proof. Let !(k) be an eigenvalue Puiseux series of L(⋅, k) expanded about k0 with limit

point !0 and � is generating eigenvector of L(⋅, k) at the point (!0, k0) and associated

with !(k). By definition of generating eigenvector, there exists a eigenpair Puiseux series

(!(k), (k)) of L(⋅, k) expanded about k0 such that !(k) has limit point !0 and (k) has

limit point �. By Lemma 59 and the fact det(J ) ∕= 0 it follows that (!(k),J (k)) is

an eigenpair Puiseux series of L∗(⋅, k) expanded about k0. Hence J (k0) = J �0 is the

limit point of the eigenvector Puiseux series J (k). By definition of generating eigenvector

this means J � is a generating eigenvector of L∗(⋅, k) at the point (!0, k0) and associated

with !(k). Moreover, since � is a generating eigenvector of L(⋅, k) at the point (!0, k0) this

151

means � ∕= 0 and � ∈ kerL(!0, k0) = ker(eik0dIn −Ψ(d, !0)). This means by Lemma 45 and

Corollary 43 that

0 ∕= q(k0,!0)(�, �) =

1

d⟨Ψ(d, !0)∗J Ψ!(d, !0)�, �⟩ℂ =

1

d⟨J Ψ!(d, !0)�,Ψ(d, !0)�⟩ℂ

=1

d⟨J Ψ!(d, !0)�, eik0d�⟩ℂ = −e

ik0d

d⟨Ψ!(d, !0)�,J �⟩ℂ.

Therefore,

⟨L!(!0, k0) 0,J �0⟩ℂ = ⟨Ψ!(d, !0)�,J �⟩ℂ ∕= 0.

This completes the proof of the lemma.

Proposition 62 If !(k) is an eigenvalue Puiseux series of L(⋅, k) expanded about k0 with

limit point !0 then !(k) is a single-valued real analytic nonconstant function of k.

Proof. Let !(k) is an eigenvalue Puiseux series of L(⋅, k) expanded about k0 with limit point

!0. Then by Lemma 60 we know that !∗(k) is also an eigenvalue Puiseux series of L(⋅, k)

expanded about k0 with limit point !0. Let !1(k) := !(k) and !2(k) := !∗(k). By definition

of the adjoint of a convergent Puiseux series we have !∗2(k) = (!∗)∗(k) = !(k) = !1(k).

Suppose that for some branch !1,ℎ1(k) of !1(k) and some branch !2,ℎ2(k) of !2(k) there

existed an r > 0 such that !1,ℎ1(k) ∕= !2,ℎ2(k) for ∣k − k0∣ < r. Then this implies by

Proposition 29 that for every generating eigenvector � of L(⋅, k) at the point (!0, k0) and

associated with !1(k) and for any generating eigenvector of L∗(⋅, k) at the point (!0, k0)(=

(!0, k0)) and associated with !∗2(k)(= !1(k)) we have

⟨L!(!0, k0)�, ⟩ℂ = 0.

But by Lemma 61 we know that J � is a generating eigenvector of L∗(⋅, k) at the point

152

(!0, k0) = (!0, k0) and associated with !∗2(k) = !1(k) and

⟨L!(!0, k0)�,J �⟩ℂ ∕= 0.

Thus we have a contradiction. This contradiction implies that there must exist an r > 0

such that if !1,ℎ1(k) is any branch of !1(k) and if !2,ℎ2(k) is any branch of !2(k) then

!1,ℎ1(k) = !2,ℎ2(k) for ∣k − k0∣ < r. Thus if we fix the branch !2,ℎ2(k) then all branches of

!1(k) are equal to !2,ℎ2(k) for ∣k−k0∣ < r. By Lemma 30 this implies !1(k) is a single-valued

analytic function of k for k ∈ B(k0, r). For the exact same reasoning this implies !2(k) is

a single-valued analytic function of k and we have !1(k) = !2(k) for all k ∈ B(k0, r). But

since !(k) = !1(k) = !2(k) = !∗(k) where !∗(k) is the adjoint of the analytic function !(k),

this implies by definition of adjoint of a Puiseux series, that !(k) = !∗(k) = !(k) for all

k ∈ B(k0, r). And so !(k) is a single-valued analytic function in B(k0, r) and is real-valued

for real k, i.e., is a real analytic function of k in B(k0, r).

To complete the proof we need only show that !(k) is a nonconstant function of k, i.e.,

!(k) = !(k0) for every k in B(k0, r) is a contradiction. By hypothesis we have !(k0) = !0 and

since !(k) is an eigenvalue Puiseux series of L(⋅, k) this means 0 = detL(!(k), k) for every

k in B(k0, r). If !(k) was a constant function then this would mean 0 = detL(!(k), k) =

detL(!0, k) = det(eikdIn − Ψ(d, !0)) for every k in B(k0, r). But det(�In − Ψ(d, !0)) is a

monic polynomial in � and so has a nonzero but finite number of zeros where as f(k) := eikd

being a nonconstant analytic function is an open map and hence f(B(k0, r)) is a nonempty

open set in ℂ and hence contains an uncountable number of elements. But then because

0 = det(f(k)In − Ψ(d, !0)) for every k in B(k0, r) this implies f(B(k0, r)) has only a finite

number of elements since they are all roots of det(�In−Ψ(d, !0)). A contradiction. Therefore

!(k) is a nonconstant function. This completes the proof.

Lemma 63 Let !1(k) and !2(k) are two different eigenvalue Puiseux series of L(⋅, k) ex-

153

panded about k0 with limit point !0. If �1, �2 are generating eigenvectors of L(⋅, k) at the

point (!0, k0) corresponding to !1(k) and !2(k), respectively, then

⟨L!(!0, k0)�1,J �2⟩ℂ = 0. (4.74)

Proof. Let !1(k) and !2(k) are two different eigenvalue Puiseux series of L(⋅, k) expanded

about k0 with limit point !0. Let �1, �2 are generating eigenvectors of L(⋅, k) at the point

(!0, k0) corresponding to !1(k) and !2(k), respectively. Then by Proposition 62 !1(k) and

!2(k) are single-valued real analytic functions and hence there adjoints satisfy !∗j(k) =

!j(k) = !j(k) for j = 1, 2. By Lemma 61 we know J �2 is a generating eigenvector of L∗(⋅, k)

at the point (!0, k0) = (!0, k0) and associated with !2(k). This implies by Proposition 29,

since !1(k) and !∗2(k) = !2(k) are different analytic functions, that

⟨L!(!0, k0)�1,J �2⟩ℂ = 0. (4.75)

This completes the proof.

Proposition 64 The holomorphic matrix function L(⋅, k0) has !0 an eigenvalue of finite

algebraic multiplicity. Furthermore, it is a semisimple eigenvalue of L(⋅, k0). Moreover, its

geometric multiplicity as an eigenvalue of L(⋅, k0) is g and this is the order of the analytic

function detL(!, k0) = D(k0, !) at ! = !0.

Proof. First, since (k0, !0) ∈ ℬℝ then 0 = D(k0, !0) = det(eik0dIn − Ψ(d, !0)) =

(−1)n det(L(!0, k0)) so that !0 ∈ �(L(⋅, k0)) and hence by definition !0 an eigenvalue of

the holomorphic matrix function L(⋅, k0). To show that it is an eigenvalue of finite algebraic

multiplicity and a semisimple eigenvalue, it suffices to show, by Proposition 15, that

L!(!0, k0) 0 = −L(!0, k0) 1

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has no solution for 0 ∈ ker(L(!0, k0))/{0} and 1 ∈ ℂn.

Suppose there existed vectors 0 ∈ ker(L(!0, k0))/{0}, 1 ∈ ℂn such that L!(!0, k0) 0 =

−L(!0, k0) 1. It follows from this, the fact ker(L(!0, k0)) = ker(eik0dIn − Ψ(d, !0)), Propo-

sition 45 along with the fact (k0, !0) is a point of definite type, that we have

0 ∕= q(k0,!0)

( 0, 0) =1

d⟨Ψ(d, !0)∗J Ψ!(d, !0) 0, 0⟩ℂ

=1

d⟨J Ψ!(d, !0) 0,Ψ(d, !0) 0⟩ℂ =

1

d⟨J Ψ!(d, !0) 0, e

ik0d 0⟩ℂ

= −eik0d

d⟨Ψ!(d, !0) 0,J 0⟩ℂ = −e

ik0d

d⟨L!(!0, k0) 0,J 0⟩ℂ.

But L(!0, k0) 0 = 0 and so by Lemma 59 this implies 0 = L∗(!0, k0)J 0 = L(!0, k0)∗J 0

which implies

⟨L(!0, k0) 1,J 0⟩ℂ = ⟨ 1, L(!0, k0)J 0⟩ℂ = 0

which implies

0 ∕= −eik0d

d⟨L!(!0, k0) 0,J 0⟩ℂ = −e

ik0d

d⟨L!(!0, k0) 0 + L(!0, k0) 1,J 0⟩ℂ = 0.

This is a contradiction. Thus !0 is an eigenvalue of L(⋅, k0) of finite algebraic multiplicity

and a semisimple eigenvalue. And so its geometric multiplicity equals its algebraic mul-

tiplicity and hence by Proposition 10, its geometric multiplicity is dim ker(L(!0, k0)) =

dim ker(eik0dIn −Ψ(d, !0)) = g and its algebraic multiplicity again g is equal to the order of

the analytic function det(L(!, k0)) at ! = !0. And since det(L(!, k0)) = (−1)n det(eik0dIn−

Ψ(d, !0)) = D(k0, !) for ! ∈ B(!0, r0) this implies g is the order of the zero of the analytic

function D(k0, !) at ! = !0. This completes the proof of the proposition.

Proposition 65 The holomorphic matrix function L(!0, ⋅) has k0 an eigenvalue of finite

algebraic multiplicity. Moreover, k0 as an eigenvalue of L(!0, ⋅) has g,m1 ≥ . . . ≥ mg,m

155

as its geometric, partial, and algebraic multiplicities, respectively. In particular, the order of

the zero of the analytic function detL(!0, k) = D(k, !0) at k = k0 is m.

Proof. We have L(!0, k) = Ψ(d, !0)−eikdIn. If we define the holomorphic matrix functions

F (�) := Ψ(d, !0)−�In and G(k) := Ψ(d, !0)− eikdIn then it follows that G(k) = F (eikd). It

thus follows that k0 is an eigenvalue of G, �0 := eik0d is an eigenvalue of F , and there is a one-

to-one correspondence with their geometric, partial, and algebraic multiplicities. But as we

know by Example 1 in chapter 3 and by definition of the numbers g, m1 ≥ . . . ≥ mg, and m,

it follows that the geometric, partial, and algebraic multiplicities of the eigenvalue �0 of F are

just, in the Jordan normal form of the matrix Ψ(d, !0) corresponding to the eigenvalue �0, g

the number of Jordan blocks, their orders m1, . . . ,mg, and m = m1 + ⋅ ⋅ ⋅+mg, respectively.

Thus g, m1, . . . ,mg, and m are the geometric, partial, and algebraic multiplicities of the

eigenvalue k0 of L(!0, k). By Proposition 10, since m being the algebraic multiplicity of the

eigenvalue k0 of L(!0, ⋅), we know that m is the order of the zero of the function detL(!0, k)

at k = k0. And since detL(!0, k) = D(k, !0), the proof is complete.

Theorem 66 There exists a � > 0 such that B(k0, �) × B(!0, �) ⊆ U and ℬ ∩ B(k0, �) ×

B(!0, �) is the union of the graphs of g nonconstant real analytic functions !1(k), . . . , !g(k)

given by convergent power series

!j(k) = !0 + �j,mj(k − k0)mj +∞∑

l=mj+1

�j,l(k − k0)l, ∣k − k0∣ < � (4.76)

where

�j,mj ∕= 0, (4.77)

for j = 1, . . . , g. Moreover, there exists analytic functions '1(k), . . . , 'g(k) belonging to

156

O(B(k0, �),ℂn) such that

'j(k) ∕= 0, L(!j(k), k)'j(k) = 0, ∣k − k0∣ < �, j = 1, . . . , g (4.78)

and the vectors '1(k), . . . , 'g(k) are linearly independent for ∣k − k0∣ < �.

Proof. The spectrum �(L) of the holomorphic matrix-valued function L is given by

�(L) = {(!, k) ∈ U ∣ detL(!, k) = 0}.

It follows from the definition of L and the set U = B(!0, r0) × B(k0, r0) ⊆ Ω × ℂ that the

inverse set �(L)−1 is a subset of the Bloch variety ℬ since

�(L)−1 = {(k, !) ∈ B(k0, r0)×B(!0, r0) ∣ detL(!, k) = 0}

= {(k, !) ∈ B(k0, r0)×B(!0, r0) ∣ detD(k, !) = 0}

= ℬ ∩B(k0, r0)×B(!0, r0).

Now by Proposition 64 the algebraic multiplicity of the eigenvalue !0 of L(⋅, k0) is g and so

by Theorem 21, there exists an � > 0, which we may assume without loss of generality � ≤ r0,

such that �(L)−1 ∩ B(k0, �) × B(!0, �) is the union of the graphs of l convergent Puiseux

series !1(k), . . . , !l(k) expanded about k0 with limit point !0, domain B(k0, �), and periods

q1, . . . , ql, respectively, which satisfy g = q1 +⋅ ⋅ ⋅+ql. This implies each !j(k) is an eigenvalue

Puiseux series of L(⋅, k) expanded about k0 with limit point !0 and so by Proposition 62,

!j(k) is a single-valued nonconstant real analytic function of k in the domain B(k0, �), for

j = 1, . . . , l. Then, since � ≤ r0 ≤ � and �(L)−1 = ℬ ∩ B(k0, r0) × B(!0, r0), we have

�(L)−1 ∩B(k0, �)×B(!0, �) = ℬ ∩B(k0, �)×B(!0, �) and it is the union of the graphs of g

convergent Puiseux series !1(k), . . . , !l(k) single-valued nonconstant real analytic function

of k in the domain B(k0, �) with !j(k0) = !0, for j = 1, . . . , l.

157

Let !1(k), . . . , !g(k) denote all those analytic functions which are not identically equal on

B(k0, �).

This implies now that they have a convergent series expansion

!j(k) = !0 + �j,qj(k − k0)qj +∞∑

l=qj+1

�j,l(k − k0)l, ∣k − k0∣ < �

where

�j,qj ∕= 0,

for j = 1, . . . , g. Without loss of generality we may assume q1 ≥ ⋅ ⋅ ⋅ ≥ qg.

Now by the Weierstrass Preparation Theorem, see Theorem 18 in the chapter 3, since detL ∈

O((!0, k0)), detL(!0, k0) = 0, detL(⋅, k0) ∕= 0 which follows since L(⋅, k0) has !0 as a

semisimple eigenvalue, then

detL = f0f1, f0 ∈ O0((!0, k0)), f1 ∈ O((!0, k0))

where f1(!, k) ∕= 0 for (!, k) ∈ B(!0, r)×B(k0, r) with 0 < r ≪ 1 and deg f0 = g, since g is

the order of the zero of detL(!, k0) = f0(!, k0)f1(!, k0) because by Proposition 64 we know

!0 is a semisimple eigenvalue of L(⋅, k0), which means its geometric multiplicity g equals

its algebraic multiplicity equals the order of the zero of detL(!, k0) at ! = !0 which thus

equals the order of the zero of f0(!, k0) at ! = !0 which is by definition the deg f0. Now by

Theorem 19 since f0 ∕= 1 because deg f0 = g > 1 there exists unique and distinct irreducibles

p1, . . . , ps ∈ O0((!0, k0)) such that

f0 =s∏j=1

pm(j)j

158

where m(j) is the unique number representing the multiplicity of the irreducible factor pj,

for j = 1, . . . , s and g = deg f0 =∑s

j=1m(j) deg pj. But for pj(!, k) := ! − !j(k) =

(! − !0)− (!j(k)− !0) we have pj ∈ O0((!0, k0)) and it irreducible, for j = 1, . . . , g. Now

though as we have shown �(L)−1 ∩ B(k0, �) × B(!0, �) is the graph of the !1(k), . . . , !g(k)

and hence since detL = f0f1 then the zeros of f0 are exactly the zeros of pj ∈ O0((!0, k0))

implying by Theorem 20 and the uniqueness of the factorization into irreducibles we must

have after possible reordering of the indices, g = s, pj = pj for j = 1, . . . , g, and

f0 =

g∏j=1

pm(j)j .

This of course implies, since deg pj = 1 for each j, that

g = deg f0 =

g∑j=1

deg pm(j)j =

g∑j=1

m(j) deg pj =

g∑j=1

m(j).

Now it follows from what we have just shown that

detL(!0, k) = f0(!0, k)f1(!0, k) = f1(!0, k)

g∏j=1

pm(j)j (!0, k)

= f1(!0, k)

g∏j=1

(!0 − !j(k))m(j) = f1(!0, k)

g∏j=1

(−�j,qj(k − k0)qj + o((k − k0)qj))m(j)

=

(f1(!0, k)

g∏j=1

(−�j,qj)m(j)

)(k − k0)

∑gj=1 qjm(j) + o

((k − k0)

∑gj=1 qjm(j)

)

=

(f1(!0, k0)

g∏j=1

(−�j,qj)m(j)

)(k − k0)

∑gj=1 qjm(j) + o

((k − k0)

∑gj=1 qjm(j)

)

as k → k0. The leading term f1(!0, k0)g∏j=1

(−�j,qj)m(j) ∕= 0 implying the order of the zero

of detL(!0, k) at k = k0 is∑g

j=1 qjm(j), but by Proposition 65 the order of the zero of

159

detL(!0, k) at k = k0 is m. This implies

m =

g∑j=1

qjm(j).

Lemma 67 For 0 < ∣k − k0∣ ≪ 1, !j(k) is a semisimple eigenvalue of L(⋅, k) and

m(j) = dim kerL(!j(k), k), j = 1, . . . , g. (4.79)

Proof. For 0 < ∣k − k0∣ ≪ 1 we have !j1(k) ∕= !j2(k) for j1 ∕= j2 with 1 ≤ j1, j2 ≤ g, since

by assumption the functions !1(k), . . . , !g(k) where not identically equal on B(k0, �). And

since

detL(!, k) = f0(!, k)f1(!, k) = f1(!, k)

g∏j=1

(! − !j(k))m(j), f1(!, k) ∕= 0

for all (!, k) ∈ B(!0, r) × B(k0, r) for 0 < r ≪ 1, these fact together with !j(k0) = !0

imply that if 0 < ∣k − k0∣ ≪ 1 then the order of the zero of the function detL(⋅, k) at

! = !j(k) is m(j) which by 10 this is the algebraic multiplicity of the eigenvalue !j(k) of

the holomorphic matrix function L(⋅, k), for j = 1, . . . , g. Now it follows by the local Smith

form (see Theorem 9) of the holomorphic matrix function L(!j(k), k) at the eigenvalue k0

that

�(j) := dim kerL(!j(k), k), j = 1, . . . , g, 0 < ∣k − k0∣ ≪ 1,

are constant. Our goal is to now show that �(j) = m(j), for j = 1, . . . , g. Notice first that by

definition of �(j), it is the geometric multiplicity of the eigenvalue of !j(k) of the holomorphic

matrix function L(⋅, k) and since m(j) is the algebraic multiplicity of the eigenvalue !j(k) of

160

the holomorphic matrix function L(⋅, k) for 0 < ∣k − k0∣ ≪ 1. But it follows from Theorem

47 that if ∣k− k0∣ ≪ 1 and k is real then (k, !j(k)) is point of definite type for the canonical

ODEs in (4.13) and hence by Proposition 64 we know !j(k) is a semisimple eigenvalue of

the holomorphic matrix function L(⋅, k) which by definition of semisimple eigenvalue means

its geometric multiplicity �(j) equals its algebraic multiplicity m(j), i.e., �(j) ≤ m(j), for

j = 1, . . . , g. And therefore this implies for all k satisfying 0 < ∣k − k0∣ ≪ 1 we have

m(j) = �(j) = dim kerL(!j(k), k), j = 1, . . . , g,

which means !j(k) is a semisimple eigenvalue of L(⋅, k). This completes the proof.

Now it follows this lemma and the local Smith form, Theorem 9, of the holomorphic matrix

function L(!j(k), k) in the variable k with eigenvalue k0, that there exists an r > 0 and there

exists function Nj,Mj ∈ O(B(k0, r),Mn(ℂ)) with Nj(k),Mj(k) invertible matrices and

Nj(k)L(!j(k), k)Mj(k) = diag{0, . . . , 0, (k − k0)mj,m(j)+1 , . . . , (k − k0)mj,n} (4.80)

for every k ∈ B(k0, r), where the number of zeros down the diagonal is given by m(j) and

{mj,i}ni=m(j)+1 is a decreasing sequence of nonnegative integers, for j = 1, . . . , g.

Let e1, . . . , en denote the standard orthonormal vectors from ℂn. We now define the vector-

valued functions

'j,i(k) := Mj(k)ei, i = 1, . . . ,m(j), j = 1, . . . , g, ∣k − k0∣ < r.

It follows, since Mj ∈ O(B(k0, r),Mn(ℂ)), that 'j,i ∈ O(B(k0, r),ℂn) for all i, j and

L(!j(k), k)'j,i(k) = 0, i = 1, . . . ,m(j), j = 1, . . . , g, ∣k − k0∣ < r.

161

Lemma 68 For every k with ∣k − k0∣ ≪ 1, the vectors 'j,i(k), i = 1, . . . ,m(j), j = 1, . . . , g

are linear independent, {'j,i(k)}m(i)i=1 is a basis for kerL(!j(k), k), for j = 1, . . . , g, and∪g

j=1{'j,i(k0)}m(i)i=1 is a basis for kerL(!0, k0).

Proof. For the first part of the theorem it suffices to prove that their limit points 'j,i(k0),

i = 1, . . . ,m(j), j = 1, . . . , g are linearly independent. Suppose they were not. Then there

exists constants cj,i, i = 1, . . . ,m(j), j = 1, . . . , g with∑g

j=1

∑m(j)i=1 ∣cj,i∣ > 0 such that

0 =

g∑j=1

m(j)∑i=1

cj,i'j,i(k0).

But if j1, j2 ∈ {1, . . . , g, i1 ∈ {1, . . . ,m(j1), and i2 ∈ {1, . . . ,m(j2)} then (!j1(k), 'i1,j1(k))

and (!j2(k), 'i2,j2(k)) are eigenpair Puiseux series of L(⋅, k) expanded about k0. By the

definition of generating eigenvector, 'j1,i1(k0) and 'j2,i2(k0) are generating eigenvectors of

L(⋅, k) at the point (!0, k0) and associated with !1(k) and !2(k), respectively. Now if j1 ∕= j2

then !j1(k) and !j2(k) are two different eigenvalue Puiseux series of L(⋅, k) expanded about

k0 with the same limit point !0. This implies by Lemma 63 that

0 = ⟨L!(!0, k0)'j1,i1(k0),J'j2,i2(k0)⟩ℂ = −⟨J Ψ!(d, !0)'j1,i1(k0), 'j2,i2(k0)⟩ℂ

= −eik0de−ik0d⟨J Ψ!(d, !0)'j1,i1(k0), 'j2,i2(k0)⟩ℂ

= −e−ik0d⟨J Ψ!(d, !0)'j1,i1(k0),Ψ(d, !0)'j2,i2(k0)⟩ℂ

= −e−ik0d⟨Ψ(d, !0)∗J Ψ!(d, !0)'j1,i1(k0), 'j2,i2(k0)⟩ℂ

= −de−ik0dq(k0,!0)

('j1,i1(k0), 'j2,i2(k0))

= −de−ik0d sgn(q(k0,d)

)⟨'j1,i1(k0), 'j2,i2(k0)⟩(k0,!0)

.

162

Thus we have the orthogonality relations

⟨'j1,i1(k0), 'j2,i2(k0)⟩(k0,!0)

= 0, j1 ∕= j2.

Now if we let j :=∑m(j)

i=1 cj,i'j,i(k0) then it follows from these orthogonality relations that

⟨ j1 , j2⟩(k0,!0) = 0, j1 ∕= j2.

And hence since 0 =∑g

j=1

∑m(j)i=1 cj,i'j,i(k0) =

∑gj=1 j it follows that

0 =

⟨g∑j=1

j, j1

⟩(k0,!0)

= ⟨ j1 , j1⟩(k0,!0) .

But by Proposition 45 we know that ⟨ , ⟩(k0,!0)

is an inner product on kerL(!0, k0) and

so we conclude j = 0 for every j. But Mj(k0) is an invertible matrix and hence 0 =

Mj(k0)−1 j =∑m(j)

i=1 cj,iei which implies that∑m(j)

i=1 ∣cj,i∣ = 0 for every j. Thus we conclude∑gj=1

∑m(j)i=1 ∣cj,i∣ = 0 a contradiction. Therefore the limit points 'j,i(k0), i = 1, . . . ,m(j),

j = 1, . . . , g are linearly independent. This completes the proof.

Without loss of generality we may take 0 < r ≤ � such that the consequences of Lemma 68 is

true for the functions 'j,i(k)), i = 1, . . . ,m(j), j = 1, . . . , g. Define the functions !j,i(k) :=

!j(k), i = 1, . . . ,m(j), j = 1, . . . , g on B(k0, r). It follows that there is∑g

j=1

∑m(j)i=1 = g of

them, they are nonconstant real analytic functions with the union of there graphs being ℬ∩

B(k0, r)×B(!0, "). The order of the zero of !j(k)−!0 at k = k0 is qj,i := qj, i = 1, . . . ,m(j),

j = 1, . . . , g such that∑g

j=1

∑m(j)i=1 qi,j =

∑gj=1m(j)qi,j = m. Moreover, the functions 'j,i(k),

i = 1, . . . ,m(j), j = 1, . . . , g are analytic functions belonging to O(B(k0, r),ℂn) such that

for each k ∈ B(k0, r) the vectors 'j,i(k), i = 1, . . . ,m(j), j = 1, . . . , g are linear independent,

{'j,i(k)}m(i)i=1 are a basis for kerL(!j,i(k), k), for j = 1, . . . , g, and

∪gj=1{'j,i(k0)}m(i)

i=1 is a

basis for kerL(!0, k0). Thus the proof this theorem will be complete if we can show that qj,i,

163

i = 1, . . . ,m(j), j = 1, . . . , g are the dimensions of the Jordan blocks in the Jordan normal

form of Ψ(d, !0) corresponding to the eigenvalue eik0d.

It follows from what we just stated that (!i,j(k), 'j,i(k)), i = 1, . . . ,m(j), j = 1, . . . , g are

eigenpair Puiseux series of L(⋅, k). And the order of the zero of !j,i(k)− !0 at k = k0 is qj,i,

i = 1, . . . ,m(j), j = 1, . . . , g. We denote any one of them by (!(k), '(k)). Then both !(k)

and '(k) are analytic at k = k0 with !(k0) = !0. Let q be the order of the zero of !j(k)−!0

at k = k0. It follows from Proposition 32 that '(k) is a generating function of order q for

L(!0, ⋅) at the eigenvalue k0. It follows by Proposition 11 that

'l :=1

j!'(l)(�0), l = 0, . . . , q − 1

is a Jordan chain of L(!0, ⋅) of length q corresponding to the eigenvalue k0. By definition of

a Jordan chain for a holomorphic function we have

'0 ∕= 0 andl∑

ℎ=0

1

ℎ!L(0,ℎ)(!0, k0)'l−ℎ = 0, l = 0, . . . , q − 1.

But L(0,0)(!0, k0) = (Ψ(d, !0) − eik0dIn) and L(0,ℎ)(!0, k0) = −(id)ℎeik0dIn for ℎ ≥ 0. Hence

for l = 0, . . . , q − 1 we have

0 = (Ψ(d, !0)− eik0dIn)'0, '0 ∕= 0,

0 =l∑

ℎ=0

1

ℎ!L(0,ℎ)(!0, k0)'l−ℎ = (Ψ(d, !0)− eik0dIn)'l +

l∑ℎ=1

−(id)ℎeik0d'l−ℎ, l ≥ 1

Thus it follow that

(Ψ(d, !0)− eik0dIn)'0 = 0, '0 ∕= 0,

(Ψ(d, !0)− eik0dIn)l'l = (ideik0d)l'0, 1 ≤ l ≤ q − 1

164

This implies that q−1 :='q−1

(ideik0d)q−1 is a generalized eigenvector of Ψ(d, !) of order q corre-

sponding to the eigenvalue eik0d. Thus if we define

l := (Ψ(d, !0)− eik0dIn)q−1−l q−1, l = 0, . . . , q − 1

then { l}q−1l=0 is a Jordan chain of Ψ(d, !0) of length q corresponding to the eigenvalue eik0d

with 0 = '0 the eigenvector.

Thus to each pair (!i,j(k), 'j,i(k)) we get a Jordan chain, { j,i,l}qj,i−1l=0 , of Ψ(d, !0) of length

qj,i−1 corresponding to the eigenvalue eik0d with j,i,0 = 'j,i(k0) the eigenvector. Thus since

the elements in a Jordan chain are always linearly independent and, since∪gj=1{'j,i(k0)}m(i)

i=1

is a basis for kerL(!0, k0), the eigenvectors of these chains j,i,0 = 'j,i(k0), i = 1, . . . ,m(j),

j = 1, . . . , g are linearly independent and this implies the collection of Jordan chains∪gj=1{'j,i(k0)}m(i)

i=1 are linearly independent. But the dimension of the generalized eigenspace

for Ψ(d, !0) corresponding to the eigenvalue eik0d is equal to its algebraic multiplicity which is

m. And the collection of Jordan chains∪gj=1{'j,i(k0)}m(i)

i=1 are linearly independent and there

are a total of∑g

j=1

∑m(j)i=1 qi,j = m of them and they belong to the generalized eigenspace

for Ψ(d, !0) corresponding to the eigenvalue eik0d. Therefore the collection of Jordan chains∪gj=1{'j,i(k0)}m(i)

i=1 are a basis for the generalized eigenspace for Ψ(d, !0) corresponding to

the eigenvalue eik0d, and in particular, they are a Jordan basis for that space for matrix

Ψ(d, !0). This implies in the Jordan normal form of Ψ(d, !0) corresponding to the eigen-

value eik0d it has∑g

j=1

∑m(j)i=1 = g Jordan blocks whose orders are exactly qi,j, i = 1, . . . ,m(j),

j = 1, . . . , g, but by hypothesis these orders where m1, . . . ,mg implying by the uniqueness

of the orders of the Jordan blocks these lists have the same numbers in them. This therefore

completes the proof.

The proof of Theorem 48.(i) now follows from Proposition 64 and Proposition 65. The

proof of Theorem 48.(ii) follows from Theorem 66. This completes the proof of Theorem 48.

165

Proof. [Corollary 49] Suppose the conditions

D(k0, !0) = 0,∂D

∂!(k0, !0) ∕= 0, (k0, !0) ∈ ℝ× Ωℝ

are satisfied where D(k, !) = det(eikdIn − Ψ(d, !)), defined in Corollary 38. By Corollary

38 the first condition implies (k0, !0) ∈ ℬℝ. Choose r > 0 such that B(!0, r) ⊆ Ω. Let

�0 := eik0d and M(") := Ψ(d, " + !0) for " ∈ B(0, r). By Proposition 33.(iii) we have

Ψ(d, ⋅) ∈ O(B(!0, r),Mn(ℂ)) and hence M(⋅) ∈ O(B(0, r),Mn(ℂ)) which implies M(") is

an analytic matrix function of the perturbation parameter ". Furthermore, �0 is an eigenvalue

of the unperturbed matrix M(0) = Ψ(d, !0) since 0 = D(!0, k0) = det(eik0dIn −Ψ(d, !0)) =

det(�0In −Ψ(d, !0)). Moreover, we have

∂

∂"det(�In −M("))∣(0,�0) =

∂

∂"det(�In −Ψ(d, "+ !0))∣(0,�0)

=∂

∂!det(�In −Ψ(d, !))∣(!0,�0) =

∂

∂!det(eikdIn −Ψ(d, !))∣(!0,k0)

=∂D

∂!(k0, !0) ∕= 0

Thus the generic condition, defined in [61, see (1.1)], is satisfied for the matrix function

M(") for the unperturbed matrix M(0) with the eigenvalue �0. By [61, Theorem 2.1.(iv)]

the Jordan normal form of M(0) = Ψ(d, !0)) corresponding to the eigenvalue �0 = eik0d

consists of a single m ×m Jordan block and there exists an eigenvector 0 of Ψ(d, !0) and

an eigenvector �0 of Ψ(d, !0)∗ corresponding to the eigenvalue �0 = e−ik0d such that

⟨Ψ!(d, !0) 0, �0⟩ℂ ∕= 0.

But since the Jordan normal form of Ψ(d, !0)) corresponding to the eigenvalue �0 = eik0d

consists of a single m × m Jordan block, this implies that dim ker(�0In − Ψ(d, !0)) = 1.

166

But because Ψ(d, !0)∗ is a adjoint of Ψ(d, !0) this implies the Jordan normal form corre-

sponding to the eigenvalue �0 = e−ik0d also consists of a single m×m Jordan block so that

dim ker(�0In−Ψ(d, !0)) = 1 as well. And therefore, since any eigenvector of Ψ(d, !0) cor-

responding to the eigenvalue �0 must be a scalar multiple of 0 and similarly any eigenvector

� of Ψ(d, !0)∗ corresponding to the eigenvalue �0 must be a scalar multiple of �0, we must

have

⟨Ψ!(d, !0) , �⟩ℂ ∕= 0

for any eigenvector of Ψ(d, !0) corresponding to the eigenvalue �0 = eik0d and for any

eigenvector � of Ψ(d, !0)∗ corresponding to the eigenvalue �0 = e−ik0d. That is the key

statement.

Indeed, Proposition 39 implies

Ψ(d, !0)∗J Ψ(d, !0) = J

and by Proposition 33.(v) we know that Ψ(d, !0) is an invertible matrix and hence for any

eigenvector of Ψ(d, !0) corresponding to the eigenvalue �0 = eik0d we have

Ψ(d, !0)∗J = J Ψ(d, !0)−1 = e−ik0dJ .

Then since J is invertible this implies J is an eigenvector of Ψ(d, !0)∗ corresponding to

the eigenvalue e−ik0d = �0. But this means

0 ∕= ⟨Ψ!(d, !0) ,J ⟩ℂ = −eik0de−ik0d⟨J Ψ!(d, !0) , ⟩ℂ

= −e−ik0d⟨J Ψ!(d, !0) ,Ψ(d, !0) ⟩ℂ

= −e−ik0d⟨Ψ(d, !0)∗J Ψ!(d, !0) , ⟩ℂ

167

= −de−ik0d 1

d

∫ d

0

⟨A!(x, !) (x), (x)⟩ℂdx, := Ψ(⋅, !0) ,

where the last equality follows from Corollary 43. Thus we have shown for any ∈

ker(eik0dIn −Ψ(d, !0)) with ∕= 0 we have

1

d

∫ d

0

⟨A!(x, !) (x), (x)⟩ℂdx ∕= 0, := Ψ(⋅, !0) .

Therefore if is any nontrivial Bloch solution of the canonical ODEs in 4.13 with (k0, !0)

as its wavenumber-frequency pair, the by Theorem 35 there exists an eigenvector of the

monodromy matrix Ψ(d, !0) corresponding to the eigenvalue eik0d such that = Ψ(⋅, !0) ,

i.e., ∈ ker(eik0dIn−Ψ(d, !0)) with ∕= 0, and thus from what we just proved we must have

1

d

∫ d

0

⟨A!(x, !) (x), (x)⟩ℂdx ∕= 0.

This proves (k0, !0) ∈ ℬℝ is a point of definite type for the canonical ODEs in 4.13. More-

over, from what we have shown the Jordan normal form of Ψ(d, !0)) corresponding to the

eigenvalue �0 = eik0d consists of a single m×m Jordan block.

We will now prove the converse. Suppose that (k0, !0) ∈ ℬℝ is a point of definite type for the

canonical ODEs in (4.13) such that the Jordan normal form of Ψ(d, !0) corresponding to the

eigenvalue �0 = eik0d consists of a single m×m Jordan block, for some m ∈ ℕ. By Theorem

48.(i), since g = 1 in this case, we know that the order of the zero of the function D(k0, !)

at ! = !0 is 1. But this is equivalent to D(k0, !0) = 0 and ∂D∂!

(k0, !0) ∕= 0. Therefore the

converse is true. This completes the proof.

168

4.5 Auxiliary Material

In this section we give some notation, convention, and background material frequently used

throughout this chapter.

Notation and Convention

(i) Mm,n(E) := the set of all m× n matrices with entries in a space E.

(ii) Mm(E) := Mm,m(E).

(iii) Em := Mm,1(E).

(iv) As convention we identify E with E1 and the set of m-tuples whose entries are in E

with M1,m(E).

(v) AT :=the transpose of the matrix A ∈Mm,n(E).

(vi) (Mm,n(E), ∣∣ ⋅ ∣∣Mm,n(E)) := the Banach space with norm

∣∣A∣∣Mm,n(E) :=

(m∑i=1

n∑i=1

∣∣aij∣∣2) 1

2

, for any A := [aij]m,ni=1,j=1 ∈Mm,n(E) (4.81)

where (E, ∣∣ ⋅ ∣∣) is a Banach space.

(vii) L (E,F ) := the Banach space of all continuous linear operators from a Banach space

(E, ∣∣ ⋅ ∣∣E) to a Banach space (F, ∣∣ ⋅ ∣∣F ) with the operator norm

∣∣T ∣∣L (E,F ) := supe∈E,e ∕=0

∣∣Te∣∣F∣∣e∣∣E

(viii) B(a, r) := {e ∈ E : ∣∣e − a∣∣ < r}, where (E, ∣∣ ⋅ ∣∣) is a Banach space, a ∈ E, and

r > 0.

169

(ix) O(Ω, E) := {f : Ω→ E ∣ f is holomorphic on Ω} where (E, ∣∣ ⋅ ∣∣) is a Banach space, Ω

an open connected set in ℂ, and f : Ω → E is said to be holomorphic on Ω provided

f is differentiable at each point in Ω, i.e., the limit

f!(!0) := lim!→!0

(! − !0)−1(f(!)− f(!0))

exists for every !0 ∈ Ω. The function f! : Ω→ E is called the derivative of f .

(x) (ℂ, ∣ ⋅ ∣):= complex numbers with the standard norm ∣a+ ib∣ :=√a2 + b2 and complex

conjugation a+ ib := a− ib, for a, b ∈ ℝ.

(xi) ∣∣ ⋅ ∣∣ℂ := ∣∣ ⋅ ∣∣Mm,n(ℂ)

(xii) Im := the identity matrix in Mm(ℂ).

(xiii) A := the matrix formed from the matrix A ∈ Mm,n(ℂ) by replacing each entry by its

complex conjugate.

(xiv) A∗ := AT

, the adjoint of the matrix A ∈Mm,n(ℂ).

(xv) ⟨u, v⟩ℂ := u∗v, the standard inner product on ℂm.

(xvi) (Lp(a, b), ∣∣ ⋅ ∣∣p) := the Banach space of measurable functions f on (a, b) (modulo

functions which vanish almost everywhere (a.e.)) such that

∣∣f ∣∣p :=

(∫ b

a

∣f(x)∣pdx) 1

p

<∞ (1 ≤ p <∞),

∣∣f ∣∣∞ := ess supx∈(a,b) ∣f(x)∣ <∞ (p =∞).

(xvii) ∣∣ ⋅ ∣∣p := ∣∣ ⋅ ∣∣Mm,n(Lp(a,b))

(xviii) Lploc(ℝ) := the space of all measurable functions f on ℝ (modulo functions which vanish

a.e.) such that f ∣(a,b) ∈ Lp(a, b) for all a < b.

170

(xix) Lp(T) := {f ∈ Lploc(ℝ) : f(x+ d) = f(x) for a.e. x ∈ ℝ}, the Banach space with norm

∣∣f ∣∣Lp(T) :=

(1

d

∫ d

0

∣f(x)∣pdx) 1

p

<∞ (1 ≤ p <∞),

∣∣f ∣∣L∞(T) := ess supx∈(0,d) ∣f(x)∣ <∞ (p =∞).

(xx) ∣∣ ⋅ ∣∣Lp(T) := ∣∣ ⋅ ∣∣Mm,n(Lp(T))

(xxi) As convention we identify an element A of Mm,n(Lp(a, b)) or Mm,n(Lploc(ℝ)) with any

one of its representative functions A(⋅) (and visa versa). And hence any identities,

inequalities, etc. throughout this chapter are to be understood as being true almost

everywhere unless otherwise stated.

(xxii) As convention we identify an element A ∈ O(Ω,Mm,n(Lp(a, b))) with the function

A : (a, b) × Ω → Mm,n(ℂ) defined by A(⋅, !) := A(!). Similarly, a function A :

(a, b)× Ω→Mm,n(ℂ) is said to be in O(Ω,Mm,n(Lp(a, b))) as a function of frequency

and we write A ∈ O(Ω,Mm,n(Lp(a, b))) provided A(!) := A(⋅, !) ∈ Mm,n(Lp(a, b)) for

each ! ∈ Ω and A(⋅) ∈ O(Ω,Mm,n(Lp(a, b))).

(xxiii) As convention we identify an element A ∈ O(Ω,Mm,n(Lp(T))) with the function A :

ℝ × Ω → Mm,n(ℂ) defined by A(⋅, !) := A(!). Similarly, a function A : ℝ × Ω →

Mm,n(ℂ) is said to be in O(Ω,Mm,n(Lp(T))) as a function of frequency and we write

A ∈ O(Ω,Mm,n(Lp(T))) provided A(!) := A(⋅, !) ∈Mm,n(Lp(T))) for each ! ∈ Ω and

A(⋅) ∈ O(Ω,Mm,n(Lp(T)))).

(xxiv)∫UA(x)dx :=

[∫Uaij(x)dx

]m,ni=1,j=1

, for any measurable set U contained in [a, b] and any

A := [aij]m,ni=1,j=1 ∈Mm,n(Lp(a, b)) ∪Mm,n(Lploc(ℝ)).

(xxv) As convention we set∫ xcA(t)dt :=

∫[c,x]

A(t)dt if x ≤ c and∫ xcA(t)dt := −

∫ cxA(t)dt if

c < a, whenever c, x ∈ [a, b] and A ∈Mm,n(Lp(a, b)) ∪Mm,n(Lploc(ℝ)).

171

(xxvi) W 1,1(a, b) := {f ∈ L1(a, b) ∣ f ′ ∈ L1(a, b)}, the standard Sobolev space with norm

∣∣f ∣∣1,1 := ∣∣f ∣∣1 + ∣∣f ′∣∣1

where f ′ denotes the weak derivative of f .

(xxvii) ∣∣ ⋅ ∣∣1,1 := ∣∣ ⋅ ∣∣Mm,n(W 1,1(a,b))

(xxviii) W 1,1loc (ℝ) := {f ∈ L1

loc(ℝ) ∣ f ′ ∈ L1loc(ℝ)}.

(xxix) A′ :=[a′ij]m,ni=1,j=1

, for any A := [aij]m,ni=1,j=1 in Mm,n(W 1,1(a, b)) or Mm,n(W 1,1

loc (ℝ)).

(xxx) As convention we always identify an element A of Mm,n(W 1,1(a, b)) or Mm,n(W 1,1loc (ℝ))

with its unique absolutely continuous or unique locally absolutely continuous repre-

sentative, respectively, (see Lemma 70 below) so that, according to this identification,

A(x) = A(u) +∫ xuA′(t)dt for all u, x ∈ [a, b], if A ∈ Mm,n(W 1,1(a, b)), and for all

u, x ∈ ℝ, if A ∈Mm,n(W 1,1loc (ℝ)).

(xxxi) As convention we identify an element A ∈ O(Ω,Mm,n(W 1,1(a, b))) with the function

A : [a, b] × Ω → Mm,n(ℂ) defined by A(x, !) := A(!)(x) – the value of the function

A : Ω→Mm,n(W 1,1(a, b)) at ! ∈ Ω and the value of the unique absolutely continuous

representative of A(!) at x ∈ [a, b]. Similarly, a function A : I × Ω → Mm,n(ℂ),

where I := (a, b) or [a, b], is said to be in O(Ω,Mm,n(W 1,1(a, b))) and we write A ∈

O(Ω,Mm,n(W 1,1(a, b))) provided A(!) := A(⋅, !) ∈ Mm,n(W 1,1(a, b)) for each ! ∈ Ω

and A(⋅) ∈ O(Ω,Mm,n(W 1,1(a, b))).

Background Material

The following is background material for this chapter.

In what follows we will need notation for the derivative of a function in the classical sense. Let

172

f : I → ℂ be a complex-valued function defined on an interval I ⊆ ℝ. If f is differentiable

(in the classical sense) everywhere on I we denote its derivative by df/dx, otherwise we

define the function df/dx : I → ℂ by

df

dx

∣∣∣x=t

:=

⎧⎨⎩ limx→tf(x)−f(t)

x−t , if the limit exists

0, else.

Lemma 69 If A ∈ Mm,n(W 1,1loc (ℝ)) then A∣(a,b) ∈ Mm,n(W 1,1(a, b)) and A∣′(a,b) = A′∣(a,b) for

any bounded interval (a, b) ⊆ ℝ.

Proof. We first prove the statement in the case m = n = 1. Suppose f ∈ W 1,1loc (ℝ). Then

by the definition of the weak derivative ()′ since f ′ ∈ L1loc(ℝ) we have

∫ℝ'′(x)f(x)dx = −

∫ℝ'(x)f ′(x)dx, ∀' ∈ C∞0 (ℝ),

where C∞0 (ℝ) is the set of all infinitely differentiable function on ℝ with compact support

and '′ = d�/dx for ' ∈ C∞0 (ℝ). This implies that for any (a, b) open interval and any

� ∈ C∞0 (a, b), the set of infinitely differentiable functions on (a, b) with compact support, we

can take the zero extension of �, i.e.,

�e(x) :=

⎧⎨⎩ �(x) if x ∈ (a, b)

0 else.

so that �e ∈ C∞0 (ℝ). And hence

∫ b

a

'′(x)f(x)dx =

∫ℝ'′e(x)f(x)dx = −

∫ℝ'e(x)f ′(x)dx

= −∫ b

a

'(x)f ′(x)dx, ∀' ∈ C∞0 (a, b).

173

This implies by the definition of the weak derivative that f ∣(a,b) ∈ W 1,1(a, b) and f ∣′(a,b) =

f ′∣(a,b). This proves the statement for m = n = 1. For the case m,n ≥ 1, it follows

now trivially from the notation 4.5.(xxix) and the fact [aij]m,ni=1,j=1∣(a,b) = [aij∣(a,b)]m,ni=1,j=1 for

[aij]m,ni=1,j=1 ∈Mm,n(L1(a, b)) ∪Mm,n(L1

loc(ℝ)). This completes the proof.

Definition 26 For any compact interval [a, b] ⊆ ℝ, denote by AC[a, b] the space of absolutely

continuous functions on [a, b], i.e.,

AC[a, b] := {f : [a, b]→ ℂ ∣ f is differentiable a.e., df/dx ∈ L1(a, b), and

f(u) = f(v) +

∫ u

v

df

dx

∣∣∣x=tdt, ∀u, v ∈ [a, b] }.

We denote by ACloc(ℝ) the space of locally absolutely continuous functions on ℝ, i.e.,

ACloc(ℝ) := {f : ℝ→ ℂ ∣ f is differentiable a.e., df/dx ∈ L1loc(ℝ), and

f(u) = f(v) +

∫ u

v

df

dx

∣∣∣x=tdt, ∀u, v ∈ ℝ }.

For A := [aij]m,ni=1,j=1 ∈ Mm,n(AC[a, b]) we define dA

dx:= [

daijdx

]m,ni=1,j=1 ∈ Mm,n(L1(a, b)). Sim-

ilarly, for A := [aij]m,ni=1,j=1 ∈ Mm,n(ACloc(ℝ)) we define the matrix dA

dx:= [

daijdx

]m,ni=1,j=1 ∈

Mm,n(L1loc(ℝ)).

Lemma 70 A ∈Mm,n(W 1,1(a, b)) if and only if A(⋅) ∈Mm,n(AC[a, b]) for some representa-

tive function A(⋅) of A. Furthermore, this representative function is unique. Moreover, for

the absolutely continuous representative A(⋅) of A we have A′ = dA(⋅)dx

and

A(x) = A(u) +

∫ x

u

A′(t)dt, ∀u, x ∈ [a, b].

Similarly, A ∈Mm,n(W 1,1loc (ℝ)) if and only if A(⋅) ∈Mm,n(ACloc(ℝ))) for some representative

function A(⋅) of A. Furthermore, this representative function is unique. Moreover, for the

174

locally absolutely continuous representative A(⋅) of A we have A′ = dA(⋅)dx

and

A(x) = A(u) +

∫ x

u

A′(t)dt, ∀u, x ∈ ℝ.

Proof. We first prove the statement for m = n = 1. By [45, p. 56, §II.2.1, Proposition

2.1.5], if f ∈ W 1,1(a, b) then f has a representative function f(⋅) : [a, b]→ ℂ such that

f(x) = f(u) +

∫ x

u

f ′(t)dt, ∀u, x ∈ [a, b].

This implies f(⋅) ∈ AC[a, b] with df(⋅)/dx = f ′. To prove uniqueness we note that if

g ∈ AC[a, b] is another representative function of f then since f(⋅) is as well we have

g(x) = f(x) for a.e. x ∈ (a, b). But then by the fact absolutely continuous functions on [a, b]

are continuous functions on [a, b] this implies g(x) = f(x) for every x ∈ [a, b]. This proves

uniqueness. On the other hand, if f(⋅) ∈ AC[a, b] then by [45, p. 55, §II.2.1, Proposition

2.1.3] we have f ∈ W 1,1(a, b).

Now suppose f ∈ W 1,1loc (ℝ). Then by Lemma 69 we have f ∣(a,b) ∈ W 1,1(a, b) for any bounded

open interval (a, b) and f ∣′(a,b) = f ′∣(a,b). Hence from what we just proved f ∣(a,b) has an

absolutely continuous representative function f ∣(a,b)(⋅) : [a, b]→ ℂ such that

f ∣(a,b)(x) = f ∣(a,b)(u) +

∫ x

u

f ∣′(a,b)(t)dt = f ∣(a,b)(u) +

∫ x

u

f ′∣(a,b)(t)dt

= f ∣(a,b)(u) +

∫ x

u

f ′(t)dt, ∀u, x ∈ [a, b].

Define a function g(⋅) : ℝ→ ℂ by

g(x) := f ∣(a,b)(x), for any bounded interval (a, b) with x ∈ (a, b).

We first show this function is well-defined. Let (a, b) and (u, v) be any two bounded open

175

interval such that I := (a, b)∩ (u, v) ∕= ∅ and let f ∣(a,b)(⋅) ∈ AC[a, b] be an absolutely contin-

uous representative function of f ∣(a,b) ∈ W 1,1(a, b) and f ∣(u,v)(⋅) ∈ AC[u, v] be an absolutely

continuous representative function of f ∣(u,v) ∈ W 1,1(u, v) . Now fix any representative func-

tion f : ℝ → ℂ of f . Then f ∣(a,b)(x) = f(x) for a.e. x ∈ [a, b] and f ∣(u,v)(x) = f(x) for

a.e. x ∈ [u, v] implying f ∣(a,b)(x) = f(x) = f ∣(u,v)(x) for a.e. x ∈ I. Hence since f ∣(a,b)(⋅)

and f ∣(u,v)(⋅) are continuous on I this implies f ∣(a,b)(x) = f ∣(u,v)(x) for all x ∈ I. This

proves g(⋅) : ℝ → ℂ is a well-defined function. From the definition of g it also proves

g∣(a,b) = f ∣(a,b) ∈ W 1,1(a, b) for any bounded open interval (a, b) and hence g = f ∈ W 1,1loc (ℝ).

It also implies that for any u, x ∈ ℝ, if we take any bounded open interval (a, b) containing

u, x, then we have

g(x) = f ∣(a,b)(x) = f ∣(a,b)(u) +

∫ x

u

f ′(t)dt = g(u) +

∫ x

u

f ′(t)dt.

Thus implying g(⋅) ∈ ACloc(ℝ) with dg(⋅)/dx = f ′. Hence we have proven f has a locally

absolutely continuous representative f(⋅) := g(⋅) ∈ ACloc(ℝ) with df(⋅)/dx = f ′ and such

that

f(x) = f(u) +

∫ x

u

f ′(t)dt, ∀u, x ∈ ℝ.

To prove uniqueness we note that if f1, f2 ∈ ACloc(ℝ) are two representative functions of f

then f1 and f2 are equal a.e. on ℝ and so since functions in ACloc(ℝ) are continuous this

implies f1 and f2 are equal everywhere on ℝ. This proves uniqueness.

Suppose now that f ∈ ACloc(ℝ). Then we have f ∈ L1loc(ℝ) and df/dx ∈ L1

loc(ℝ). We need

only show now that f has df/dx as its weak derivative, i.e., f ′ = df/dx. To do this we need

to show

∫ℝ'′(t)f(t)dt = −

∫ℝ'(t)

df

dx

∣∣∣x=tdt, ∀' ∈ C∞0 (ℝ),

176

where C∞0 (ℝ) is the set of all infinitely differentiable function on ℝ with compact support.

Let ' ∈ C∞0 (ℝ) and let [a, b] be any nonempty compact interval such that supp(�) ⊆ [a, b].

Then by integration by parts we have

∫ℝ'′(t)f(t)dt =

∫ b

a

'′(t)f(t)dt = f(b)'(b)− f(a)'(a)−∫ b

a

'(t)df

dx

∣∣∣x=tdt

= −∫ b

a

'(t)df

dx

∣∣∣x=tdt = −

∫ℝ'(t)

df

dx

∣∣∣x=tdt.

And therefore we have shown

∫ℝ'′(t)f(t)dt = −

∫ℝ'(t)

df

dx

∣∣∣x=tdt, ∀' ∈ C∞0 (ℝ),

which proves f ′ = df/dx ∈ L1loc(ℝ) and hence f ∈ W 1,1

loc (ℝ).

This completes the prove in the case m = n = 1. The case m,n ≥ 1 follows now trivially

from the case m = n = 1 by the notation 4.5.(xxiv) and 4.5.(xxix). This completes the

proof.

The following lemma, along with Lemma 69 and Lemma 70, completes our characterization

of spaces Mm,n(W 1,1(a, b)) and Mm,n(W 1,1loc (ℝ)).

Lemma 71 Let A : ℝ → Mm,n(ℂ) be a function such that there exists an increasing se-

quence of intervals {(aj, bj)}∞j=1 with ℝ =∪∞j=1(aj, bj) such that for every j, A∣(aj ,bj) ∈

Mm,n(W 1,1(aj, bj)). Then A ∈Mm,n(W 1,1loc (ℝ)).

Proof. We first prove the statement for m = n = 1. Let f : ℝ → ℂ be a function

such that there exists an increasing sequence of intervals {(aj, bj)}∞j=1 with ℝ =∪∞j=1(aj, bj)

such that for every j, f ∣(aj ,bj) ∈ W 1,1(aj, bj). It follows from this that on every interval

(aj, bj) the function f on this interval is equal a.e. to a function which is measurable and

integrable on (aj, bj) and this implies the function f restricted to (aj, bj) is a measurable and

177

integrable function. Thus since ℝ =∪∞j=1(aj, bj) and the restriction of f : ℝ→ ℂ to (aj, bj)

is measurable and integrable for every j, this implies f is a measurable function which is

locally integrable and so f ∈ L1loc(ℝ).

Now f ∣(aj ,bj) ∈ W 1,1(aj, bj) for every j and so by Lemma 70 there exists an absolutely

continuous function gj ∈ AC[aj, bj] such that f(x) = gj(x) for a.e. x ∈ (aj, bj) for each j.

But this implies for any fix j0 we must have gj(x) = f(x) = gj0(x) for a.e. x ∈ (aj0 , bj0) for any

fixed j ≥ j0. But the functions gj, gj0 are continuous on [aj0 , bj0 ] and so this implies gj(x) =

gj0(x) for all x ∈ [aj0 , bj0 ] and every j ≥ j0. But this implies g(x) := limj→∞ gj(x) exists for

every x ∈ ℝ and so defines a function g : ℝ → ℂ. Moreover, from what we proved we have

g(x) = gj(x) for all x ∈ [aj0 , bj0 ] and any j. Thus for every j we have g∣[aj ,bj ] = gj ∈ AC[aj, bj]

implying g is differentiable a.e. on [aj, bj] with dg/dx∣[aj ,bj ] = dgj/dx = g′j ∈ L1(aj, bj). But

since ℝ =∪∞j=1(aj, bj) this implies g is differentiable a.e. on ℝ with dg/dx ∈ L1

loc(ℝ). Hence

if ' ∈ C∞0 (ℝ) then we can find j sufficiently large so that supp(') ⊆ [aj, bj] and so

∫ℝ'′(t)g(t)dt =

∫ bj

aj

'′(t)g(t)dt =

∫ bj

aj

'′(t)gj(t)dt =

∫ bj

aj

'(t)g′j(t)dt

=

∫ bj

aj

'(t)dg

dx

∣∣∣x=tdt =

∫ℝ'(t)

dg

dx

∣∣∣x=tdt.

This implies that g′ = dg/dx ∈ L1loc(ℝ) and therefore g ∈ W 1,1

loc (ℝ). But f = g a.e. on ℝ and

so f ∈ W 1,1loc (ℝ). This proves the statement for n = m = 1. The case where m,n ≥ 1 follows

trivially now from the case n = m = 1. This completes the proof.

Lemma 72 Let (a, b) ⊆ ℝ be a bounded interval. Then the matrix multiplication ⋅ :

Mm,n(W 1,1(a, b)) × Mn,r(W1,1(a, b)) → Mm,r(W

1,1(a, b)), where A ⋅ B := AB for A ∈

Mm,n(W 1,1(a, b)), B ∈Mn,r(W1,1(a, b)), is a continuous bilinear map and

(AB)′ = A′B + AB′.

178

Proof. We first prove the statement for m = n = r = 1. To begin, let (a, b) ⊆ ℝ

be a bounded interval. Then by [45, p. 65, §II.2.3, Proposition 2.3.1], multiplication ⋅ :

W 1,1(a, b) ×W 1,1(a, b) → W 1,1(a, b), where f ⋅ g := fg for f, g ∈ W 1,1(a, b), is a continuous

bilinear map and

(fg)′ = f ′g + fg′.

This proves the statement for m = n = r = 1. The statement for m,n, r ≥ 1 now follows

from this, Lemma 77, and notation 4.5.(xxix). This completes the proof.

Lemma 73 Matrix multiplication ⋅ : Mm,n(W 1,1loc (ℝ)) ×Mn,r(W

1,1loc (ℝ)) → Mm,r(W

1,1loc (ℝ)),

where A ⋅B := AB for A ∈Mm,n(W 1,1loc (ℝ)), B ∈Mn,r(W

1,1loc (ℝ)), is a well-defined map and

(AB)′ = A′B + AB′.

Proof. We first prove the statement for m = n = r = 1. To begin, let (a, b) ⊆ ℝ be any

bounded interval. It follows from Lemma 69 and Lemma 72 that if f, g ∈ W 1,1loc (ℝ) then we

have (fg)∣(a,b) = f ∣(a,b)g∣(a,b) ∈ W 1,1(a, b). Now since this is true for every bounded interval

(a, b) ⊆ ℝ, this implies by Lemma 71 that fg ∈ W 1,1loc (ℝ). Moreover, by Lemma 69 and

Lemma 72 we have for every bounded interval (a, b) ⊆ ℝ,

((fg)∣(a,b))′ = (f ∣(a,b)g∣(a,b))′ = (f ∣(a,b))′g∣(a,b) + f ∣(a,b)(g∣(a,b))′

= f ′∣(a,b)g∣(a,b) + f ∣(a,b)g′∣(a,b).

It follows from this that f ′g + fg′ ∈ L1loc(ℝ) and for any ' ∈ C∞0 (ℝ) we can find a bounded

interval (a, b) ⊆ ℝ such that supp(') ⊆ (a, b) implying

∫ℝ'′(x)(fg)(x)dx =

∫ b

a

'′(x)(fg)∣(a,b)(x)dx

179

= −∫ b

a

'(x)(f ′∣(a,b)(x)g∣(a,b)(x) + f ∣(a,b)(x)g′∣(a,b)(x))dx

= −∫ b

a

'(x)(f ′(x)g(x) + f(x)g′(x))dx

= −∫ b

a

'(x)(f ′g + fg′)(x)dx.

And because this is true for every ' ∈ C∞0 (ℝ) this means the weak derivative of fg is

f ′g + fg′, i.e.,

(fg)′ = f ′g + fg′.

This proves the statement for m = n = r = 1. The statement for m,n, r ≥ 1 now follows

trivially from this one by notation 4.5.(xxix). This completes the proof.

Lemma 74 Let E = Mm,n(Lploc(ℝ)) or Mm,n(W 1,1loc (ℝ)), where 1 ≤ p ≤ ∞. The translation

operator ℒd : E → E, where (ℒdA)(⋅) = A(⋅+d) for A ∈ E, is a well-defined map. Moreover,

for any A ∈Mm,n(W 1,1loc (ℝ)) we have

(ℒdA)′ = ℒdA′.

Proof. We first prove the statement for n = m = 1. Let f ∈ Lploc(ℝ). Then (ℒdf)(⋅) : ℝ→

ℂ given by (ℒdf)(x) := f(x + d), x ∈ ℝ is a measurable function. And for any bounded

interval (a, b) ⊆ ℝ, since ∣∣(ℒdf)∣(a,b)∣∣p = ∣∣f ∣(a+d,b+d)∣∣p, we have (ℒdf)∣(a,b) ∈ Lp(a, b). This

proves ℒdf ∈ Lploc(ℝ). Now if f ∈ W 1,1loc (ℝ) then f ′ ∈ L1

loc(ℝ) and from what we have just

proved, ℒdf ′ ∈ L1loc(ℝ). We will now show ℒdf ∈ W 1,1

loc (ℝ) with (ℒdf)′ = ℒdf ′. To start, we

have ('(⋅−d))′ = '′(⋅−d) ∈ C∞0 (ℝ) for all ' ∈ C∞0 (ℝ) since ()′ = d/dx is just the derivative

in the classical sense on smooth functions. From this and the fact the Lebesgue integral on

180

ℝ is translation invariant it follows that

∫ℝ�′(x)(ℒdf)(x)dx =

∫ℝ�′(x− d)(ℒdf)(x− d)dx

=

∫ℝ�′(x− d)f(x)dx =

∫ℝ�′(x− d)f(x)dx

=

∫ℝ(�(⋅ − d))′(x)f(x)dx =

∫ℝ�(x− d)f ′(x)dx

=

∫ℝ�(x)f ′(x+ d)dx =

∫ℝ�(x)(ℒdf ′)(x)dx, ' ∈ C∞0 (ℝ).

But this implies (ℒdf)′ = ℒdf ′ and, since we have already shown ℒdf ′ ∈ L1loc(ℝ), this proves

ℒdf ∈ W 1,1loc (ℝ). This proves the statement for m = n = 1. The statement for m,n, r ≥ 1

now follows trivially from this one by notation 4.5.(xxix). This completes the proof.

Lemma 75 For 1 ≤ p ≤ ∞ and m,n ∈ ℕ, the adjoint extends via A∗(⋅) := A(⋅)∗ for

A ∈ Mm,n(Lploc(ℝ)), to a well-defined map ∗ : Mm,n(Lploc(ℝ))→ Mn,m(Lploc(ℝ)). Moreover, if

A ∈Mm,n(W 1,1loc (ℝ)) then A∗ ∈Mn,m(W 1,1

loc (ℝ)) and

(A∗)′ = (A′)∗.

Proof. In the case m = n = 1, it just follows from the facts that f ∈ L1loc(ℝ) if and only

if Re(f), Im(f) ∈ L1loc(ℝ) and f ∈ W 1,1

loc (ℝ) if and only if Re(f), Im(f) ∈ W 1,1loc (ℝ), Re(f ′) =

(Re(f))′, Im(f ′) = (Im(f))′, and f ′ = f′. In the case m,n ≥ 1, if A := {aij}m,ni=1,j=1 ∈

Mm,n(L1loc(ℝ)) then, by the case m = n = 1, we have A∗ = {a∗ij}

n,mj=1,i=1 ∈Mn,m(L1

loc(ℝ)) and

if A ∈Mm,n(W 1,1loc (ℝ)) then A∗ ∈Mn,m(W 1,1

loc (ℝ)) with

(A∗)′ = {aij ′}n,mj=1,i=1 ={aij ′}n,mj=1,i=1

= (A′)∗.

This completes the proof.

181

Lemma 76 If A ∈ Mm,n(Lp(T)) and 1 ≤ p ≤ ∞ then A∗ ∈ Mn,m(Lp(T)) with ∣∣A∗∣∣Lp(T) =

∣∣A∣∣Lp(T).

Proof. By Lemma 75 the map ∗ : Mm,n(Lploc(ℝ))→Mn,m(Lploc(ℝ)) extending the adjoint by

A∗(⋅) = A(⋅)∗ for A ∈Mm,n(Lploc(ℝ)) is well-defined. By Lemma 74, the translation operator

ℒd : Mm,n(Lploc(ℝ)) → Mm,n(Lploc(ℝ)) given by (LdA)(⋅) = A(⋅ + d) for A ∈ Mm,n(Lploc(ℝ))

is well-defined. Thus for any A ∈ Mm,n(Lp(T)), since Mm,n(Lp(T)) ⊆ Mm,n(Lploc(ℝ)), then

A∗,ℒdA∗, (ℒdA)∗ are all well-defined and belong to Mn,m(Lploc(ℝ)). Also, B ∈ Mn,m(Lp(T))

if and only if Mn,m(Lploc(ℝ)) and ℒdB = B. Thus since A∗ ∈Mn,m(Lploc(ℝ)) and

ℒdA∗ = A∗(⋅+ d) = A(⋅+ d)∗ = (ℒdA)∗ = A∗

then A∗ ∈Mn,m(Lp(T)).

Now we prove ∣∣A∗∣∣Lp(T) = ∣∣A∣∣Lp(T) for A ∈Mm,n(Lp(T)). We start with the case m = n =

1. If f ∈ Lp(T) then as we just proved f ∈ Lp(T) and

∣∣f ∣∣Lp(T) =

(1

d

∫ d

0

∣f(x)∣pdx) 1

p

=

(1

d

∫ d

0

∣f(x)∣pdx) 1

p

= ∣∣f ∣∣Lp(T) (1 ≤ p <∞),

∣∣f ∣∣L∞(T) = ∣∣f ∣∣∞ = ∣∣f ∣∣∞ = ∣∣f ∣∣L∞(T) (p =∞).

This proves the statement for m = n = 1. Consider now the case m,n ≥ 1. If A :=

{aij}m,ni=1,j=1 ∈Mm,n(Lp(T)) then A∗ = {aij}n,mj=1,i=1 ∈Mn,m(Lploc(ℝ)) and so

∣∣A∗∣∣Lp(T) =

(n∑j=1

m∑i=1

∣∣aij∣∣2Lp(T)

) 12

=

(m∑i=1

n∑j=1

∣∣aij∣∣2Lp(T)

) 12

= ∣∣A∣∣Lp(T).

And hence the statement is true for m,n ≥ 1. This completes the proof.

182

General Statement

In this section we give the statements we need for general Banach spaces. We split this

section into statements that do not involve the frequency ! and those that do.

Parameter Independent

This is the collection of statements which do not involve the frequency parameter.

Definition 27 Let E1, E2, and E3 be Banach spaces. A function ⋅ : E1×E2 → E3 is called

a continuous bilinear map provided it satisfies the following three properties:

(i) For any a1 ∈ E1, the map a2 7→ a1 ⋅ a2 is a linear map from E2 to E3.

(ii) For any a2 ∈ E2, the map a1 7→ a1 ⋅ a2 is a linear map from E1 to E3.

(iii) There exists a constant C > 0 such that ∣∣a1 ⋅ a2∣∣E3 ≤ C∣∣a1∣∣E1∣∣a2∣∣E2 for all a1 ∈ E1,

a2 ∈ E2.

Lemma 77 Let E1, E2, and E3 be Banach spaces and ⋅ : E1 × E2 → E3 be a bilinear map.

Define the map ⋅ : Mm,n(E1)×Mn,r(E2)→Mm,r(E3) defined by

A ⋅B :=

[n∑k=1

aik ⋅ bkj

]m,ri=1,j=1

for any A := [aij]m,ni=1,j=1 ∈ Mm,n(E1), B := [bij]

n,ri=1,j=1 ∈ Mn,r(E2). Then ⋅ : Mm,n(E1) ×

Mn,r(E2)→Mm,r(E3) is a continuous bilinear map.

Proof. The map is obviously linear in both of its arguments and so properties (i) and (ii)

are satisfied of Definition 27. Now choose any C > 0 such that ∣∣a1 ⋅a2∣∣E3 ≤ C∣∣a1∣∣E1∣∣a2∣∣E2

183

for all a1 ∈ E1, a2 ∈ E2. Then

∣∣A ⋅B∣∣E3 =

⎛⎝ m∑i=1

r∑j=1

∥∥∥∥∥n∑k=1

aik ⋅ bkj

∥∥∥∥∥2

E3

⎞⎠ 12

≤

(m∑i=1

r∑j=1

n∑k=1

∣∣aik ⋅ bkj∣∣2E3

) 12

≤

(m∑i=1

r∑j=1

n∑k=1

C2∣∣aik∣∣2E1∣∣bkj∣∣2E3

) 12

≤ C

((m∑i=1

n∑k=1

∣∣aik∣∣2E1

)(n∑k=1

r∑j=1

∣∣bkj∣∣2E3

)) 12

= C∣∣A∣∣Mm,n(E1)∣∣B∣∣Mn,r(E2).

This proves property (iii) of Definition 27 and thus we have proved the lemma.

Parameter Dependent

This is the collection of statements which involve the frequency parameter.

Lemma 78 Let E1, E2, and E3 be Banach spaces and ⋅ : E1 × E2 → E3 be a bilinear map.

If u ∈ O(Ω, E1) and v ∈ O(Ω, E2) then u ⋅ v ∈ O(Ω, E3) and

(u ⋅ v)! = u! ⋅ v + u ⋅ v!.

Proof. First we choose any C > 0 such that ∣∣a1 ⋅ a2∣∣E3 ≤ C∣∣a1∣∣E1∣∣a2∣∣E2 for all a1 ∈ E1,

a2 ∈ E2. Now let u ∈ O(Ω, E1) and v ∈ O(Ω, E2). Let !0 ∈ Ω. Then

∥∥(! − !0)−1(u(!) ⋅ v(!)− u(!0) ⋅ v(!0))− (u!(!0) ⋅ v(!0) + u(!0) ⋅ v!(!0))∥∥E3

=∥∥[(! − !0)−1(u(!)− u!(!0))] ⋅ v(!) + u(!0) ⋅ [(! − !0)−1(v(!)− v!(!0))]

∥∥E3

≤ C∣∣(! − !0)−1(u(!)− u!(!0))∣∣E1∣∣(! − !0)−1(v(!)− v!(!0))∣∣E2

184

Thus as ! → !0 the RHS of that inequality goes to zero implying the LHS of that inequality

does as well which implies the function (u ⋅ v)(!) := u(!) ⋅ v(!) is differentiable at !0 and

(u ⋅ v)!(!0) = (u! ⋅ v)(!0) + (u ⋅ v!)(!0).

Therefore u ⋅ v ∈ O(Ω, E3) and (u ⋅ v)! = u! ⋅ v + u ⋅ v!.

Lemma 79 Let E1 and E2 be Banach spaces. If ℒ ∈ O(Ω,L (E1, E2)) and u ∈ O(Ω, E1)

then ℒu ∈ O(Ω, E2) and

(ℒu)! = ℒ!u+ ℒu!.

Proof. From the Banach spaces L (E1, E2) and E1, we define the map ⋅ : L (E1, E2)×E1 →

E2 by ℒ ⋅u := ℒu. Then it is a continuous bilinear map with ∣∣ℒ ⋅u∣∣E2 ≤ ∣∣ℒ∣∣L (E1,E2)∣∣u∣∣E1 .

Thus it follows from Lemma 78 above that if ℒ ∈ O(Ω,L (E1, E2)) and u ∈ O(Ω, E1) then

ℒu ∈ O(Ω, E2) and (ℒu)! = ℒ!u+ ℒu!. This completes the proof.

Lemma 80 Let E be a Banach space. Then O(Ω,Mm,n(E)) = Mm,n(O(Ω, E)) and for any

A := [aij]m,ni=1,j=1 ∈ O(Ω,Mm,n(E)),

A! = [(aij)!]m,ni=1,j=1.

Proof. Let (E, ∣∣⋅∣∣) be a Banach space and (Mm,n(E), ∣∣⋅∣∣Mm,n(E)) the Banach space whose

norm is defined in (4.81). To begin, we know that any two norms on a finite dimensional

vector space over ℂ are equivalent so that there exists constants C1, C2 > 0 such that

C1∣∣B∣∣Mm,n(ℂ),∞ ≤ ∣∣B∣∣Mm,n(ℂ),2 ≤ C2∣∣B∣∣Mm,n(ℂ),∞, ∀B ∈Mm,n(ℂ),

185

where ∣∣ ⋅ ∣∣Mm,n(ℂ),∞, ∣∣ ⋅ ∣∣Mm,n(ℂ),2 : Mm,n(ℂ)→ [0,∞) are the norms defined by

∣∣B∣∣Mm,n(ℂ),∞ := max1≤i≤m,1≤j≤n

∣bij∣, ∣∣B∣∣Mm,n(ℂ),2 :=

(m∑i=1

n∑i=1

∣bij∣2) 1

2

,

for any B := [bij]m,ni=1,j=1 ∈Mm,n(ℂ).

Now we define a function ∣ ⋅ ∣ : Mm,n(E)→ [0,∞) by

∣A∣ := [∣∣aij∣∣]m,ni=1,j=1, ∀A := [aij]m,ni=1,j=1 ∈Mm,n(E).

We define two norms ∣∣ ⋅ ∣∣Mm,n(E),∞, ∣∣ ⋅ ∣∣Mm,n(E),2 : Mm,n(E)→ [0,∞) by

∣∣A∣∣Mm,n(E),∞ :=∣∣∣∣∣A∣∣∣∣∣

Mm,n(ℂ),∞, ∣∣A∣∣Mm,n(E),2 :=∣∣∣∣∣A∣∣∣∣∣

Mm,n(ℂ),2

for any A := [aij]m,ni=1,j=1 ∈Mm,n(E). But since ∣∣ ⋅ ∣∣Mm,n(E),2 = ∣∣ ⋅ ∣∣Mm,n(E) it follows that

C1∣∣A∣∣Mm,n(E),∞ ≤ ∣∣A∣∣Mm,n(E) ≤ C2∣∣A∣∣Mm,n(E),∞, ∀A ∈Mm,n(E).

From this it follows that the identity map � : (Mm,n(E), ∣∣ ⋅ ∣∣Mm,n(E)) → (Mm,n(E), ∣∣ ⋅

∣∣Mm,n(E),∞) belongs to L ((Mm,n(E), ∣∣ ⋅ ∣∣Mm,n(E)), (Mm,n(E), ∣∣ ⋅ ∣∣Mm,n(E),∞)) and has a con-

tinuous inverse. These facts and Lemma 79 imply that O(Ω, (Mm,n(E), ∣∣ ⋅ ∣∣Mm,n(E))) =

O(Ω, (Mm,n(E), ∣∣ ⋅ ∣∣Mm,n(E),∞)) and (�A)! = �A! = A! for all A ∈ O(Ω, (Mm,n(E), ∣∣ ⋅

∣∣Mm,n(E))).

Now it follows from the definition of the norm ∣∣ ⋅ ∣∣Mm,n(E),∞ that the statement of this

lemma is true for O(Ω, (Mm,n(E), ∣∣ ⋅ ∣∣Mm,n(E),∞)) and hence O(Ω, (Mm,n(E), ∣∣ ⋅ ∣∣Mm,n(E))) =

O(Ω, (Mm,n(E), ∣∣ ⋅ ∣∣Mm,n(E),∞)) = Mm,n(O(Ω, E)) with

[(aij)!]m,ni=1,j=1 = (�A)! = �A! = A!

186

for every A := [aij]m,ni=1,j=1 ∈ O(Ω, (Mm,n(E), ∣∣ ⋅ ∣∣Mm,n(E))). This completes the proof of the

lemma.

Lemma 81 If f ∈ O(Ω, E) then f! ∈ O(Ω, E).

Proof. This statement is proven in [19, pp. 21–22, Theorem 1.8.4 & Theorem 1.8.5].

Specific Statements

In the second section we give the statements we need for the specific Banach spaces used in

this chapter. We split this section into statements that do not involve the frequency ! and

those that do.

Parameter Independent

This is the collection of statements which do not involve the frequency parameter.

Lemma 82 If p, q, s ∈ ℝ with 1 ≤ p, q, s ≤ ∞ and 1p

+ 1q

= 1s

then matrix multiplication ⋅ :

Mm,n(Lp(a, b))×Mn,r(Lq(a, b))→Mm,r(L

s(a, b)), where A⋅B := AB for A ∈Mm,n(Lp(a, b)),

B ∈Mn,r(Lq(a, b)), is a continuous bilinear map.

Proof. Multiplication ⋅ : Lp(a, b)× Lq(a, b)→ Ls(a, b), where f ⋅ g := fg for f ∈ Lp(a, b),

g ∈ Ls(a, b), is a bilinear map since it follows from Holder’s inequality that

∣∣fg∣∣s ≤ ∣∣f ∣∣p∣∣g∣∣q.

187

Thus by Lemma 77 it follows that matrix multiplication ⋅ : Mm,n(Lp(a, b))×Mn,r(Lq(a, b))→

Mm,r(Ls(a, b)), where A ⋅ B := AB for A ∈ Mm,n(Lp(a, b)) and B ∈ Mn,r(L

q(a, b)), is also

continuous bilinear map. This completes the proof.

Lemma 83 Let 1 ≤ p ≤ ∞. Then the map of matrix multiplication ⋅ : Mm,n(Lp(a, b)) ×

Mn,r(W1,1(a, b)) → Mm,r(L

p(a, b)), where A ⋅ B := AB for A ∈ Mm,n(Lp(a, b)) and B ∈

Mn,r(W1,1(a, b)), is a continuous bilinear map.

Proof. Let (C[a, b], ∣∣ ⋅ ∣∣∞) denote the Banach space of all continuous functions on [a, b]

with norm ∣∣g∣∣∞ := supx∈[a,b] ∣g(x)∣. By our convention, if g ∈ W 1,1(a, b) then g is absolutely

continuous on [a, b] and hence g ∈ C[a, b]. Thus we have W 1,1(a, b) ⊆ C[a, b]. Now by [45,

p. 57, §II.2.1, Proposition 2.1.7] we have for the inclusion map � : W 1,1(a, b)→ C[a, b] given

by �g = g that � ∈ L (W 1,1(a, b), C[a, b]). This implies there exists a C > 0 such that

supx∈[a,b]

∣g(x)∣ = ∣∣g∣∣∞ ≤ C∣∣g∣∣1,1, for every g ∈ W 1,1(a, b).

It follows from this that for all f ∈ Lp(a, b), g ∈ W 1,1(a, b) we have

∣∣fg∣∣p ≤ ∣∣f ∣∣p∣∣g∣∣∞ ≤ C∣∣f ∣∣p∣∣g∣∣1,1.

This implies that multiplication ⋅ : Lp(a, b) ×W 1,1(a, b) → Lp(a, b), where f ⋅ g := fg for

f ∈ Lp(a, b), g ∈ W 1,1(a, b), is a continuous bilinear map. Therefore it follows by Lemma 77

that the matrix multiplication ⋅ : Mm,n(Lp(a, b))×Mn,r(W1,1(a, b))→Mm,r(L

p(a, b)), where

A ⋅B := AB for A ∈Mm,n(Lp(a, b)), B ∈Mn,r(W1,1(a, b)), is also a continuous bilinear map.

This completes the proof.

Lemma 84 Let 1 ≤ p ≤ ∞. Then the map of matrix multiplication ⋅ : Mm,n(Lploc(ℝ)) ×

Mn,r(W1,1loc (ℝ)) → Mm,r(L

ploc(ℝ)), where A ⋅ B := AB for A ∈ Mm,n(Lploc(ℝ)) and B ∈

Mn,r(W1,1loc (ℝ)), is a well-defined map.

188

Proof. Let A ∈ Mm,n(Lploc(ℝ)) and B ∈ Mn,r(W1,1loc (ℝ)). Then, for any bounded interval

(a, b) ⊆ ℝ by Lemma 83, (AB)∣(a,b) = A∣(a,b)B∣(a,b) ∈ Mm,r(Lp(a, b)). But this implies

AB ∈Mm,r(Lploc(ℝ)). This completes the proof.

Lemma 85 For each x ∈ [a, b], the evaluation map at x defined by �x : Mm,n(W 1,1(a, b))→

Mm,n(ℂ) where �xA := A(x) for A ∈Mm,n(W 1,1(a, b)). Then

�x ∈ L (Mm,n(W 1,1(a, b)),Mm,n(ℂ)). (4.82)

Proof. It follows from the proof of Lemma 83 that there exists a constant C > 0 such that

∣�xg∣ ≤ C∣∣g∣∣1,1, for every g ∈ W 1,1(a, b).

Now the evaluation map is linear so we need only prove its bounded. If A := [aij]m,ni=1,j=1 ∈

Mm,n(W 1,1(a, b)) we have

∣∣�xA∣∣ℂ = ∣∣A(x)∣∣ℂ =

(m∑i=1

n∑j=1

∣aij(x)∣2) 1

2

≤ m12n

12 max

1≤i≤m1≤j≤n

∣aij(x)∣

≤ Cm12n

12 max

1≤i≤m1≤j≤n

∣∣aij∣∣1,1 ≤ Cm12n

12

(m∑i=1

n∑j=1

∣∣aij∣∣21,1

) 12

= Cm12n

12 ∣∣A∣∣1,1.

This implies that ∣∣�x∣∣L (Mm,n(W 1,1(a,b)),Mm,n(ℂ)) ≤ Cm12n

12 and hence it follows that �x ∈

L (Mm,n(W 1,1(a, b)),ℂ). This completes the proof.

Lemma 86 Let a, b ∈ ℝ with a < b and let u ∈ [a, b]. We define the integral map by

(ℐA)(x) :=

∫ x

u

A(t)dt, A ∈Mm,n(L1(a, b)), x ∈ (a, b).

189

Then ℐ ∈ L (Mm,n(L1(a, b)),Mm,n(W 1,1(a, b))) and

(ℐA)′ = A,

for any A ∈Mm,n(L1(a, b)).

Proof. By [45, p. 56, §II.2.1, Proposition 2.1.5] it follows that the map ℐ : Mm,n(L1(a, b))→

Mm,n(W 1,1(a, b)) is a well-defined map which is linear and (ℐA)′ = A. Thus we need only

prove that it is bounded to complete the proof. Now if f ∈ L1(a, b) then it follows from the

definition of the norm ∣∣ ⋅ ∣∣1,1 on W 1,1(a, b) that

∣∣ℐf ∣∣1,1 = ∣∣(ℐf)′∣∣1 + ∣∣ℐf ∣∣1 = ∣∣f ∣∣1 + ∣∣ℐf ∣∣1 ≤ (b− a+ 1)∣∣f ∣∣1.

Hence it follows that if A := [aij]m,ni=1,j=1 ∈Mm,n(L1(a, b)) then

∣∣ℐA∣∣1,1 =

(m∑i=1

n∑j=1

∣∣ℐaij∣∣21,1

) 12

≤ m12n

12 max

1≤i≤m1≤j≤n

∣∣ℐaij∣∣1,1

≤ (b− a+ 1)m12n

12 max

1≤i≤m1≤j≤n

∣∣aij∣∣1 ≤ (b− a+ 1)m12n

12

(m∑i=1

n∑j=1

∣∣aij∣∣21

) 12

= (b− a+ 1)m12n

12 ∣∣A∣∣1.

This implies ∣∣ℐ∣∣L (Mm,n(L1(a,b)),Mm,n(W 1,1(a,b))) ≤ (b − a + 1)m12n

12 and hence it follows that

ℐ ∈ L (Mm,n(L1(a, b)),Mm,n(W 1,1(a, b))). This completes the proof.

Lemma 87 The integral map ℐ : Mm,n(L1loc(ℝ))→Mm,n(W 1,1

loc (ℝ)) defined by

(ℐA)(x) :=

∫ x

0

A(t)dt, A ∈Mm,n(L1loc(ℝ)), x ∈ ℝ,

190

is well-defined and satisfies

(ℐA)′ = A,

for any A ∈Mm,n(L1loc(ℝ)).

Proof. Let A ∈ Mm,n(L1loc(ℝ)). Then it follows that the function BA(⋅) : ℝ → Mm,n(ℂ),

defined by

BA(x) :=

∫ x

0

A(t)dt, x ∈ ℝ,

is well-defined. Moreover, by Lemma 86 we have that BA∣(a,b) ∈ Mm,n(W 1,1loc (a, b)) for any

bounded interval (a, b) in ℝ containing 0 and BA∣′(a,b) = A∣(a,b). By Lemma 71 it follows

that BA ∈ Mm,n(W 1,1loc (ℝ)) and BA

′∣(a,b) = BA∣′(a,b) = A∣(a,b) for any bounded interval (a, b)

in ℝ containing 0. This proves B′A = A. But by the definition of the integral map we have

ℐA = BA for any A ∈ Mm,n(L1loc(ℝ)). Therefore ℐ : Mm,n(L1

loc(ℝ)) → Mm,n(W 1,1loc (ℝ)) is

well-defined and (ℐA)′ = A for any A ∈Mm,n(L1loc(ℝ)). This completes the proof.

Lemma 88 Let 1 ≤ p ≤ ∞ and (a, b) ⊆ ℝ be a bounded open interval. Define the restriction

map by A 7→ A∣(a,b) where A ∈Mm,n(Lp(T)). Then ∣(a,b) ∈ L (Mm,n(Lp(T)),Mm,n(Lp(a, b)).

Proof. Fix p ∈ [1,∞] and a bounded open interval (a, b). Pick q ∈ ℕ such that (a, b) ⊆

[−qd, qd]. Let f ∈ Lp(T). Then f ∈ Lploc(ℝ) and f(x + d) = f(x) for a.e. x ∈ ℝ. If p = ∞

then

∣∣f ∣(a,b)∣∣∞ ≤ ∣∣f ∣(−qd,qd)∣∣∞ = ∣∣f ∣(0,d)∣∣∞ = ∣∣f ∣∣L∞(T).

191

If p ∈ [0,∞) then by periodicity we have∫ jd

(j−1)d∣f(x)∣pdx =

∫ d0∣f(x)∣pdx for any j ∈ ℤ.

Hence implying

∣∣f ∣(a,b)∣∣pp =

∫ b

a

∣f(x)∣pdx ≤∫ qd

−qd∣f(x)∣pdx

=

q∑j=1

∫ jd

(j−1)d

∣f(x)∣pdx+

q∑j=1

∫ −(j−1)d

−jd∣f(x)∣pdx

=

q∑j=1

∫ d

0

∣f(x)∣pdx+

q∑j=1

∫ d

0

∣f(x)∣pdx

= 2q

∫ d

0

∣f(x)∣pdx = 2qd∣∣f ∣∣pLp(T).

Thus for any p ∈ [1,∞] we have ∣∣f ∣(a,b)∣∣p ≤ (2qd)1p ∣∣f ∣∣Lp(T). But then for any A :=

[aij]m,ni=1,j=1 ∈Mm,n(Lp(T)) we have

∣∣A∣(a,b)∣∣p =

(m∑i=1

n∑j=1

∣∣aij∣(a,b)∣∣2p

) 12

≤ m12n

12 max

1≤i≤m1≤j≤n

∣∣aij∣(a,b)∣∣p

≤ (2qd)1pm

12n

12 max

1≤i≤m1≤j≤n

∣∣aij∣∣Lp(T)

≤ (2qd)1pm

12n

12

(m∑i=1

n∑j=1

∣∣aij∣∣2Lp(T)

) 12

= (2qd)1pm

12n

12 ∣∣A∣∣Lp(T).

And hence we conclude the restriction map ∣(a,b) : Mm,n(Lp(T)) → Mm,n(Lp(a, b)) is a

bounded linear operator with norm bounded above by (2qd)1pm

12n

12 . This proves the state-

ment for p ∈ [1,∞].

Parameter Dependent

This is the collection of statements which involve the frequency parameter.

Lemma 89 Let p, q, s ∈ ℝ with 1 ≤ p, q, s ≤ ∞ and 1p

+ 1q

= 1s. If A ∈ O(Ω,Mm,n(Lp(a, b)))

192

and B ∈ O(Ω,Mn,r(Lq(a, b))) then AB ∈ O(Ω,Mm,r(L

s(a, b))) and

(AB)! = A!B + AB!.

Proof. By Lemma 82 we know that the map of matrix multiplication ⋅ : Mm,n(Lp(a, b))×

Mn,r(Lq(a, b)) → Mm,r(L

s(a, b)), where A ⋅ B := AB for A ∈ Mm,n(Lp(a, b)) and B ∈

Mn,r(Lq(a, b)), is a continuous bilinear map. Thus by Lemma 78 we must have AB ∈

O(Ω,Mm,r(Ls(a, b))) and

(AB)! = A!B + AB!.

This completes the proof.

Lemma 90 Let 1 ≤ p ≤ ∞. If A ∈ O(Ω,Mm,n(Lp(a, b))) and B ∈ O(Ω,Mn,r(W1,1(a, b)))

then AB ∈ O(Ω,Mn,r(Lp(a, b))) and

(AB)! = A!B + AB!.

Proof. By Lemma 83 we know that the map of matrix multiplication ⋅ : Mm,n(Lp(a, b))×

Mn,r(W1,1(a, b)) → Mm,r(L

p(a, b)), where A ⋅ B := AB for A ∈ Mm,n(Lp(a, b)), B ∈

Mn,r(W1,1(a, b)), is a continuous bilinear map. Thus by Lemma 78 we must have AB ∈

O(Ω,Mm,r(Lp(a, b))) and

(AB)! = A!B + AB!.

This completes the proof.

Lemma 91 Let x ∈ [a, b] and �x : Mm,n(W 1,1(a, b)) → Mm,n(ℂ) be the evaluation map

193

at x, i.e., �xA = A(x) for all A ∈ Mm,n(W 1,1(a, b)). If A ∈ O(Ω,Mm,n(W 1,1(a, b))) then

�xA ∈ O(Ω,Mm,n(ℂ)) and

(�xA)! = �xA!.

Proof. By Lemma 85 we know �x ∈ L (Mm,n(W 1,1(a, b)),Mm,n(ℂ)). Thus by Lemma 79,

if A ∈ O(Ω,Mm,n(W 1,1(a, b))) then �xA ∈ O(Ω,Mm,n(ℂ)) and

(�xA)! = �xA!.

This completes the proof.

Lemma 92 Let a, b ∈ ℝ, a < b, and u ∈ [a, b]. Let ℐ : Mm,n(L1(a, b)) → Mm,n(W 1,1(a, b))

be the integral map, i.e.,

(ℐA)(x) :=

∫ x

u

A(t)dt, A ∈Mm,n(L1(a, b)), x ∈ (a, b).

If A ∈ O(Ω,Mm,n(L1(a, b))) then ℐA ∈ O(Ω,Mm,n(W 1,1(a, b))) and

(ℐA)! = ℐA!.

Proof. By Lemma 86 we know that ℐ ∈ L (Mm,n(L1(a, b)),Mm,n(W 1,1(a, b))). Thus by

Lemma 79, if A ∈ O(Ω,Mm,n(L1(a, b))) then ℐA ∈ O(Ω,Mm,n(W 1,1(a, b))) and

(ℐA)! = ℐA!.

This completes the proof.

194

Lemma 93 Let p ∈ [1,∞) and (a, b) be any bounded open interval in ℝ. Let ∣(a,b) :

Mm,n(Lp(T))→Mm,n(Lp(a, b)) be the restriction map, i.e., the domain restriction map A 7→

A∣(a,b) for A ∈ Mm,n(Lp(T)). If A ∈ O(Ω,Mm,n(Lp(T))) then A∣(a,b) ∈ O(Ω,Mm,n(Lp(a, b)))

and

(A∣(a,b))! = A!∣(a,b).

Proof. By Lemma 88 we know that ∣(a,b) ∈ L (Mm,n(Lp(T)),Mm,n(Lp(a, b)). Thus by

Lemma 79, if A ∈ O(Ω,Mm,n(Mm,n(Lp(T)))) then we have A∣(a,b) ∈ O(Ω,Mm,n(Lp(a, b)))

and

(A∣(a,b))! = A!∣(a,b).

This completes the proof.

195

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