gkn theory for linear hamiltonian systems

Applied Mathematics and Computation 182 (2006) 1514–1527

www.elsevier.com/locate/amc

GKN theory for linear Hamiltonian systems q

Zhaowen Zheng a,*, Shaozhu Chen b

a Department of Mathematical Science, Qufu Normal University, Qufu 273165, Shandong, PR Chinab Department of Mathematical Science, Shandong University at Weihai, Weihai 263000, Shandong, PR China

Abstract

In this paper, we give the definition of maximal and minimal operators for linear Hamiltonian systems and investigatethe relationship between the conjugate scalar product in a weighted Hilbert space and the skew-symmetric boundary formof the associated singular Hamiltonian operator, namely, the one-to-one correspondence between the set of self-adjointextensions of the minimal operator and the set of Lagrangian symplectic subspaces. These results extend and improvethe classical Glazman–Krein–Naimark (GKN) theory for quasi-differential operators.� 2006 Elsevier Inc. All rights reserved.

Keywords: Hamiltonian system; Self-adjoint extension; Operator; GKN theory

1. Introduction

The spectrum of singular differential operators is related to many disparate and deep topics in mathematicsand physics, including quantum mechanics, stability of fluid and inverse problems in partial differential equa-tions, and draws attention to many new problems. Many interesting results in this field have been given forvarious differential operators (see [8–12] for details). Boundary value problems for quasi-differential equation

0096-3

doi:10

q ThResear

* CoE-m

MA½y� ¼ kwy; ð1Þ
where MA ¼ iny½n�A is a complex formal quasi-differential operator, y ½n�A is the quasi-derivative, A = An·n(t) is aShin–Zettl matrix (see [1,2] for details) defined on an assigned real interval I ¼ ða; bÞ � R with a prescribedpositive weight function w(x) > 0 and k 2 C is a complex spectral parameter, are customarily treated withinthe fundamental theory of unbounded linear operators, usually self-adjoint on some appropriate complexfunction Hilbert space, using the classical Weyl–Kodaira–Titchmarsh spectrum theory. Spectral propertiesof the operator are obtained by using the Green function and asymptotic properties of solutions to (1).
003/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

.1016/j.amc.2006.05.041

is research was partially supported by the NSF of China (Grant 10471077), NSF of Shandong (Grant Y2005A06) and the Sciencech Foundation of Shandong University at Weihai (Grant XZ 2004004).rresponding author.ail address: [email protected] (Z. Zheng).

mailto:[email protected]

Z. Zheng, S. Chen / Applied Mathematics and Computation 182 (2006) 1514–1527 1515

Looking at this problem from a different aspect, Glazman, Krein and Naimark founded a theory ofalgebraic descriptions of the set of all self-adjoint operators generated by the differential system. This theoryis now abbreviated by GKN theory to commemorate these three former Soviet mathematicians. The origi-nal GKN theory was confined to real-valued quasi-differential expression of arbitrary even order. It wasextended to complex-valued quasi-differential expression of arbitrary order earlier by Halperin [5], Shin[6] and later by Everitt [7] by introducing methods of linear complex symplectic geometry and the Lagrang-ian algebra. Indeed, it is a nontrivial extension of the theory for real differential expressions. More concisely,the key kernel of GKN theory is depicting all the self-adjoint operators with suitable boundary conditions.Firstly, one needs to define the maximal operator T1 and the minimal operator T0 by MA in a Hilbert spaceL2

wðIÞ. Secondly, GKN theory determines all the self-adjoint extensions of T0 by imposing on DðT 1Þ bound-ary conditions at the left and right end-points of I. Finally, the number of self-adjoint extensions of T0 isequal to that of Lagrangian symplectic subspaces. Moreover, the GKN theory gives an algebraic isomor-phism between the set of Lagrangian complex symplectic subspaces and the set of self-adjoint operatorsin the Hilbert space. In paper [4], Everitt and Zettl extend the GKN theory to the spectral theory for qua-si-differential expressions of arbitrary order with a countable number of intervals on the real line. For exam-ple, it includes the specific case of second order Schrodinger equations with a potential which has countablyisolated singularities.

Eq. (1) may be written as a linear Hamiltonian system using the following substitution. Let y be a solutionof (1), set

x ¼

y

y0

..

.

yðn�1Þ

0BBBB@1CCCCA; u ¼

ð�1Þn�1ðrnyðnÞÞðn�1Þ þ � � � þ r1y 0

..

.

�ðrnyðnÞÞ0 þ rn�1yðn�1Þ

rnyðnÞ

0BBBBB@

1CCCCCA:

Then the pair of n-vectors (x,u) solves the linear Hamiltonian system

x0 ¼ AðtÞxþ BðtÞu;u0 ¼ ðCðtÞ � kwðtÞE11Þx� A�ðtÞu;

�
where
BðtÞ ¼ diag 0; . . . ; 0;1

rnðtÞ

� �; CðtÞ ¼ diagfr0ðtÞ; . . . ; rn�1ðtÞg;

A ¼ ðaijÞn�n; aij ¼1 if j ¼ iþ 1; i ¼ 1; . . . ; n� 1;

0 otherwise;

(

E11 ¼ ðeijÞn�n; eij ¼1 if i ¼ j ¼ 1;

0 otherwise:

(
However, there has been little literature involving the theory of maximal and minimal operators and the
GKN theory for linear Hamiltonian systems. In this paper, we will consider a general type of Hamiltoniansystems and give the definitions of the maximal and minimal operators in Section 3. In Section 4, we willexamine the geometric and algebraic structure of all the self-adjoint extensions of the minimal operatorand develop a corresponding version of the GKN theory for linear Hamiltonian systems.

2. Symplectic spaces and Lagrangian subspaces

Definition 2.1. A complex linear space S, together with a complex-valued function on the product space S · S

S � S ! C; X ; Y ! ½X : Y �
is a pre-symplectic space in case

1516 Z. Zheng, S. Chen / Applied Mathematics and Computation 182 (2006) 1514–1527

(1) Sesquilinear, semi-linear, or conjugate bilinear property

½Z : X þ Y � ¼ ½Z : X � þ ½Z : Y �;½X þ Y : Z� ¼ ½X : Z� þ ½Y : Z�;½lX : Y � ¼ l½X : Y �; ½x : lY � ¼ l½X : Y �

for all X, Y, Z 2 S and l 2 C.(2) Skew-Hermitian or alternating property

½X : Y � ¼ �½Y : X �; 8X ; Y 2 S:

If in addition to the properties (1) and (2) we further require(3) Non-degeneracy property

8Y 2 S; ½X : Y � ¼ 0 ) X ¼ 0;

then S, together with the non-degenerate, skew-Hermitian, sesquilinear form [:], is a symplectic space, [:] is thesymplectic form of S.

Since the complex symplectic space S is similar to an Hermitian inner-product space formally, so we define[:] as a symplectic inner-product. If [X:Y] = 0, we call X, Y as symplectic orthogonal. Note that a linear sub-space of symplectic space might not be a symplectic subspace, since the non-degeneracy property might not besatisfied.

Definition 2.2. Let (S, [:]) be a pre-symplectic space. A linear subspace L � S is called Lagrangian in case

½X : Y � ¼ 0; 8X ; Y 2 L:

Moreover, a Lagrange subspace L of S is called completed provided

X 2 S; ½X : S� ¼ 0 ) X 2 L:

In a pre-symplectic space S, with the semi-bilinear form [:], we note that

½X : X � ¼ �½X : X �; so R½X : X � ¼ 0:

Also,

½lX : lX � ¼ ll½X : X � ¼ jlj2½X : X �
for each vector X 2 S, and scalar l 2 C. Hence, each vector X 2 S is of exactly one of the following threetypes:
1. positive, I½X : X � > 0;2. negative, I½X : X � < 0;3. neutral, I½X : X � ¼ 0, so [X :X] = 0.

Note that a Lagrangian subspace L � S consists of neutral vectors, with the additional property that[X:Y] = 0 for all X,Y 2 L.

3. Maximal and minimal operators

Consider the boundary value problems for linear Hamiltonian system

x0 ¼ AðtÞxþ BðtÞu;u0 ¼ ðCðtÞ � kC0ðtÞÞx� A�ðtÞu

�ð2Þ

or equivalently,

ly :¼ Jy 0 �HðtÞy ¼ kW ðtÞy; ð3Þ


where y ¼ xu

� �, HðtÞ ¼ �CðtÞ A�ðtÞ

AðtÞ BðtÞ

� �, W ðtÞ ¼ CkðtÞ 0

0 0

� �2n�2n

, C0ðtÞ ¼CkðtÞ 0

0 0

� �n�n

, J ¼

0 �In

In 0

� �is a skew-symplectic matrix, t 2 I ¼ ða; bÞ, �1 6 a < b 61, A(t), B(t), C(t), C0(t) are n · ncom-

plex valued matrices defined on I, B(t), C(t) and Ck(t) are Hermitian matrices, Ck(t) is positive definite of rankk. We call l formally Hamiltonian operator in its domain

DðlÞ ¼ y : I! C2n j y 2 AClocðIÞ� �

;

where AClocðIÞ denotes the set of all 2n-dimensional vector functions defined on I, which are local absolutelycontinuous. The linear Hamiltonian system (2) is called controllable on I if for any vector-valued solution

y ¼ xu

� �of (2), we have that if x(t) 0 on any non-degenerate subinterval I1 � I, then x(t) = u(t) 0

on I. In what follows, we always assume system (2) is controllable. We call 2n · 1 vector-valued functionf(t) is ‘‘integral square’’ provided

RI

f �ðtÞW ðtÞf ðtÞdt <1, the collections of such functions are denoted byL2

W ðIÞ. Since W(t) P 0 is semi-positive definite,

kf kW ¼ZI

f �ðtÞW ðtÞf ðtÞdt� 1=2

is a semi-norm defined on L2W ðIÞ. However, if we introduce the equivalence

y1 y2 () ky1 � y2kW ¼ 0;

then the quotient space L2W ðIÞ= is a Hilbert space. We still denote the quotient space by L2

W ðIÞ for conve-nience. The inner product in this Hilbert space is

ðf ; gÞW ¼ZI

g�ðtÞW ðtÞf ðtÞdt:

Denote the operator generated by the formally Hamiltonian operator l by T : DðT Þ ! DðlÞ; Ty ¼ f , where

DðT Þ ¼ fy 2 DðlÞ j there exists f 2 DðlÞ; such that ly ¼ Wf g:
Now, if f is a solution of system (2), then Tf = kf, and this means that f is an eigenvector of T corresponding tok. In this case, f and Tf must be in the same Hilbert space. Here, we select the Hilbert space to be L2
W ðIÞ.For simplicity, let eW ¼ diagfIk; 0g and construct a transformation U from L2

W ðIÞ to L2eW ðIÞ as follows:

8y 2 L2W ðIÞ; U ½y�ðtÞ ¼ ey ¼ diag R; In�k; In�k;R�1

�yðtÞ; ð4Þ

where R(t) = Ck(t)1/2. If R1(t) = diag(R(t), In�k), then

U ½y1�ðtÞ;U ½y2�ðtÞ �

L2eW ðIÞ ¼ZI

ey 2ðtÞ� eW ey 1ðtÞdt ¼ZI

y�2ðtÞR1ðtÞ 0

0 R�11 ðtÞ

� � eW RðtÞ 0

0 R�1ðtÞ

� �y1 dt

¼ZI

y�2ðtÞW ðtÞy1ðtÞdt ¼ y1; y2ð ÞL2W ðIÞ

:

So U is a linear isometric projection from L2W ðIÞ to L2eW ðIÞ, and U�1 exists,

U�1

½ey � ðtÞ ¼ R�11 ðtÞ 0

0 R1ðtÞ

!eyðtÞ:
For each operator T generalized by l on L2
W ðIÞ, we define an operator eT ¼ UTU�1 on L2eW ðIÞ, such that eT isgeneralized by el
ely :¼ Jy 0 � fHy
on L2eW ðIÞ, where

fH ¼ �R�11 CR1 �R�1

1 ðR01 þ A�R1ÞðR01 þ R1AÞR�1

1 R1BR1

!¼:

�eC eA�eA eB !

:


Since there exists a unitary equivalence between L2W ðIÞ and L2eW ðIÞ, the self-adjointness of T on L2

W ðIÞimplies the self-adjointness of eT on L2eW ðIÞ and vice versa. The self-adjoint boundary conditions of el can

be obtained from the self-adjoint boundary conditions of l. In fact, suppose l fulfills left boundary conditions(without loss of generality, we may suppose a = 0 and the left boundary is regular), ay(0) = 0,a = (a1,a2), i.e.,

a1x(0) + a2u(0) = 0, where a1,a2 satisfy a1a�2 þ a2a�1 ¼ 0; a1a�1 þ a2a�2 ¼ In, and ranka1

a2

� �¼ n. Sinceey ¼ U ½y� ¼

R1 00 R�1

1

� �xu

� �, we have b1exð0Þ þ b2euð0Þ ¼ 0, where b1 ¼ a1R�1

1 ð0Þ, b2 = a2R1(0). Obviously,

b1, b2 satisfy the left self-adjoint conditions, and the left boundary conditions of el are self-adjoint. Thus,

the following conclusion is immediate.

Proposition 3.1. Suppose operators l and el are defined as above. Then we have the following facts:

(1) rðlÞ ¼ rðelÞ,(2) rpðlÞ ¼ rpðelÞ,(3) reðlÞ ¼ reðelÞ.
In view of this proposition, we will always use the weight matrix eW for the linear Hamiltonian system (2).
Definition 3.2 [3, Definition 2.1.3]. A vector-valued function y ¼ xu

� �is called admissible (or (A,B)-

admissible) on I, if y 2 DðlÞ and x 0 = A(t)x + B(t)u on I for some u. Operator l is called formally self-adjointprovided

ðlf ; gÞW ¼ ðf ; lgÞW ; 8f ; g 2 D0ðlÞ;

where D0ðlÞ ¼ fy 2 DðlÞ j suppy lies in a compact set interior to Ig:

Lemma 3.3. If all the functions in D0ðlÞ are admissible, then l is formally self-adjoint.

Proof. For every f ; g 2 D0ðlÞ, suppose f ¼ f1

f2

� �, g ¼ g1

g2

� �, f 01 ¼ Af 1 þ Bf 2, g01 ¼ Ag1 þ Bg2. Now, let

½a; b� � I be large enough such that supp f, suppg � [a,b]. Then we have

ðlf ; gÞW ¼Z b

ag�W ðlf Þ ¼

Z b

ag�ðJf 0 �Hf Þ ¼

Z b

aðg�1; g�2Þ

0 �I

I 0

� �f 01f 02

� ��C A�

A B

� �f1

f2

� �� ¼Z b

a�g�1f 02 þ g�1Cf 1 � g�1A�f2 ¼ �g�1f2jba þ

Z b

ag�1Cf 1 þ g�2Bf 2 ¼

Z b

ag�1Cf 1 þ g�2Bf 2:

In the same way, we get

ðf ; lgÞW ¼Z b

ag�1Cf 1 þ g�2Bf 2:

Consequently, (lf,g)W = (f, lg)W, i.e., l is formally self-adjoint. h

By Lemma 3.3, we obtain the following corollary.

Corollary 3.4. If f ; g 2 DðlÞ are admissible, then for any compact interval ½a; b� � I, we have
Z b
ag�W ðlf Þ � ðlgÞ�Wf ¼ g�2f1 � g�1f2

� ba¼ g�Jf½ �ba :

Among all the operators on the weighted Hilbert space L2W ðIÞ, generated by the formally self-adjoint Ham-

iltonian operator l, one operator H is maximal in the sense that it requires only f and Hf lies in the Hilbertspace L2

W ðIÞ at the same time. In other words, the domain is the maximum feasible region of the boundaryvalue problems.


Definition 3.5. The operator H defined by

H : DðHÞ ! L2W ðIÞ;

y 7! Hy ¼ fð5Þ

is called the maximal operator generated by the formal Hamiltonian operator l, where

DðHÞ ¼ fy 2 AClocðIÞ \ L2W ðIÞ j 9 f 2 L2

W ðIÞ s:t: ly ¼ Wf g:

Remark 3.6. All the elements of DðHÞ are admissible.

Remark 3.7. Since the weight function is W ¼ Ik 00 0

� �2n�2n

and C0 ¼Ik 00 0

� �n�n

, we can write DðHÞ asfollows:

DðHÞ ¼ y ¼x

u

� �� x ¼ x1

x2

� �; x1 2 L2ðIÞ; 9f ¼

f1

0

� �; f 1 2 L2ðIÞ;

s:t: x0 ¼ Axþ Bu; u0 ¼ Cx� A�uþ f where x1; f1 are vectors with dimension k�:

Definition 3.8. A mapping [:] with domain DðHÞ �DðHÞ and range C is called a boundary form, if for allf ; g 2 DðHÞ,

½f : g� ¼ZI

g�W ðHf Þ � ðHgÞ�Wf ¼ ðHf ; gÞW � ðf ;HgÞW :

It is clear that boundary form is skew-symmetric, i.e., ½f : g� ¼ �½g : f � for all f ; g 2 DðHÞ:

By Remark 3.6 and Corollary 3.4, for all f ; g 2 DðHÞ and for any non-degenerate compact interval½a; b� � I, we have the Green formula:
Z b
ag�W ðHf Þ � ðHgÞ�Wf ¼ ½g�2f1 � g�1f2�ba ¼ g�Jf½ �ba : ð6Þ

When the endpoint of [a,b] tends to the endpoint a,b of interval I,

½f : g� ¼ lima!ab!b

½g�2f1 � g�1f2�ðbÞ � ½g�2f1 � g�1f2�ðaÞ� �

:

Obviously, for all f ; g 2 DðHÞ, the one-side limit exists and is finite, i.e.,

lima!a½g�2f1 � g�1f2�ðaÞ ¼ ½g�2f1 � g�1f2�ðaÞ;

limb!b½g�2f1 � g�1f2�ðbÞ ¼ ½g�2f1 � g�1f2�ðbÞ:

Now, we define a subset of L2W ðIÞ as follows:

D0ðHÞ ¼ fy 2 DðHÞ j supp y lies in a compact set interior to Ig:
We have D0ðHÞ ¼ DðHÞ \D0ðlÞ, and for all f ; g 2 D0ðHÞ, [f :g] = 0 by Corollary 3.4.
Theorem 3.9. (Density property). D0ðHÞ ¼ L2W ðIÞ.

Proof. For any non-degenerate compact interval ½a; b� � I, we define

D0ð½a; b�Þ ¼ fy 2 DðHÞ j suppy � ½a; b�g:
Obviously, D0ð½a; b�Þ � D0ðHÞ. Define
L2W ð½a; b�Þ ¼ fz 2 L2

W ðIÞ j z ¼ 0 a:e: outside ½a; b�g:


Then L2W ð½a; b�Þ is a Hilbert space (in the sense of quotient topology). It is sufficient to prove

D0ð½a; b�Þ ¼ L2W ð½a; b�Þ: ð7Þ

For this purpose, we first give the following lemma.

Lemma 3.10. For any non-degenerate compact interval ½a; b� � I,

L2W ð½a; b�Þ ¼ R0ð½a; b�Þ � P ð½a; b�Þ; ð8Þ

where R0ð½a; b�Þ � L2W ð½a; b�Þ is the range of the maximal operator H restricted to the domain D0ð½a; b�Þ;

Pð½a; b�Þ � L2W ð½a; b�Þ consists of all the solutions z(t) of the Hamiltonian system

lz ¼ Jz0 �Hz ¼ 0 on t 2 ½a; b�; and zðtÞ 0 for t 2 I n ½a; b�:

Proof. It is sufficient to verify that R0([a,b]) and P([a,b]) are closed subspace of L2W ð½a; b�Þ, and

R0ð½a; b�Þ ¼ P ð½a; b�Þ?. By Green formula, for each pair of functions f ; g 2 DðHÞ,
Z b
ag�W ðHf Þ � ðHgÞ�Wf ¼ ½g�2f1 � g�1f2�ba :

If g 2 D0ð½a; b�Þ, then ½g�2f1 � g�1f2�ðtÞ ¼ 0 for t 2 I n ½a; b�, and
Z b
ag�W ðHf Þ ¼

Z b

aðHgÞ�Wf :

For each function f 2 R0([a,b]), there exists y 2 D0ð½a; b�Þ, such that ly = Wf. Now, we prove that f is orthog-onal to P([a,b]) in L2

W ð½a; b�Þ. Let z(t) be any solution of the linear Hamiltonian system lz ¼ Jz0 �HðtÞz ¼ 0 on[a,b], we extend z(t) for t 2 I (denoted by z(t) for convenience). Then either z 2 D0ðHÞ, or z 2 P([a,b]). Wecan compute in this two cases that
Z b
az�Wf ¼

Z b

az�W ðHyÞ ¼

Z b

aðHzÞ�Wy þ z�2y1 � z�1y2

� ba:

Since Hz = 0 for t 2 [a,b] and y 2 D0ð½a; b�Þ, we obtain ½z�2y1 � z�1y2�ðaÞ ¼ ½z�2y1 � z�1y2�ðbÞ ¼ 0. Hence,R ba z�Wf ¼ 0 for all f 2 R0([a,b]), which implies f 2 P ð½a; b�Þ?, i.e., R0ð½a; b�Þ � P ð½a; b�Þ?. On the other hand,

for each f 2 P ð½a; b�Þ? � L2W ð½a; b�Þ, we show that f 2 R0([a,b]). For this purpose, fix a basis z1,z2, . . . ,z2n of

linear Hamiltonian system lz ¼ Jz0 �Hz ¼ 0 on [a,b], with the prescribed initial conditionsat t = b : (z1,z2, . . . ,z2n)(b) = I2n. We extend zv (v = 1,2, . . . , 2n) as zeros outside [a,b] (denoted these func-tions as unchangeable for convenience), so zv 2 P([a,b]). Then our assumption on f 2 P ð½a; b�Þ? guaranteesthat
Z b
az�vWf ¼ 0; v ¼ 1; 2; . . . ; 2n:

Now, let yðtÞðt 2 ½a; b�Þ be the unique solution of linear Hamiltonian system

Jy 0 �HðtÞy ¼ Wf ðtÞ; t 2 ½a; b�
with the initial condition yðaÞ ¼ 0. For any extension of yðtÞ such that yðtÞ 2 D0ðHÞ and zv(t) 2 P([a,b]), wehave
0 ¼Z b

az�vWf ¼

Z b

az�vW ðHyÞ ¼

Z b

aðHzÞ�Wy þ z�2y1 � z�1y2

� ba:

Since Hzv = 0, t 2 [a,b] and yðaÞ ¼ 0, we have z�2y1 � z�1y2

� ðbÞ ¼ 0, i.e., fz�vJ ygðbÞ ¼ 0, v = 1,2, . . . , 2n, which

implies yðbÞ ¼ 0. This implies that every extension of yðtÞ from [a,b] to I with yðtÞ 2 D0ð½a; b�Þ necessarilysatisfies yðaÞ ¼ yðbÞ ¼ 0. Therefore, f ¼ Hy 2 R0ð½a; b�Þ. So we obtain P ð½a; b�Þ? � R0ð½a; b�Þ. Now, we obtain


that (8) is true. Since the dimension of P([a,b]) is finite, we obtain P([a,b]) is also a closed subspace, its orthog-onal complement space R0([a,b]) is a closed subspace. The proof of Lemma 3.10 completes. h

With Lemma 3.10, we now prove Theorem 3.9. It is sufficient to prove that for each compact interval[a,b] interior to I, D0ð½a; b�Þ is dense in L2

W ð½a; b�Þ. Accordingly, fix such a compact interval [a,b] interiorto I, let f 2 L2

W ð½a; b�Þ satisfying

ðf ; yÞW ¼Z b

ay�Wf ¼ 0 for all y 2 D0ð½a; b�Þ:

We show that this implies

f 0; t 2 ½a; b�:
Suppose zðtÞ is any solution of Hamiltonian system Jz0 �Hz ¼ Wf for t 2 [a,b], and extend zðtÞ over I suchthat zðtÞ 2 D0ðHÞ. For each y 2 D0ð½a; b�Þ,
0 ¼ ðf ; yÞW ¼Z b

ay�Wf ¼

Z b

ay�W ðHzÞ ¼

Z b

aðHyÞ�W z:

Particularly, let zðtÞ 0, t 2 I n ½a; b�, by Lemma 3.10, z 2 R0ð½a; b�Þ? ¼ P ð½a; b�Þ. So Wf = 0 for t 2 [a,b],which implies f1 0 for t 2 [a,b] (f1 are the first k-components of h). Since system (2) is controllable, f2 0for t 2 [a,b] (f2 are the last (n � k)-components of f). So f 0. This completes the proof. h

By Theorem 3.9, l generates a symmetric operator on D0ðHÞ, we give the definition of minimal operator forlinear Hamiltonian system (2) as follows.

Definition 3.11. The operator h defined by

h : DðhÞ ! L2W ðIÞ

y 7! hy ¼ fð9Þ

is called the minimal operator generated by the formal Hamiltonian operator l, where DðhÞ ¼ fy 2DðHÞj½y : DðHÞ� ¼ 0g.

Theorem 3.12. We denote by H0, the operator generated by l on D0ðHÞ. Then

h� ¼ H ; H � ¼ h; H 0 ¼ h:

Proof. It is clear that D0ðHÞ � DðhÞ � DðHÞ. Since D0ðHÞ ¼ L2W ðIÞ, all the operators H �0, h*, H* exist and are

closed operators in L2W ðIÞ, and DðH �Þ � Dðh�Þ � D0ðH �Þ. Firstly, we prove

H �0 ¼ H :

For each f 2 D0ðH �Þ, there exists F 2 L2W ðIÞ such that

ðf ;H 0gÞW ¼ ðF ; gÞW ; 8g 2 D0ðHÞ ¼ DðH 0Þ:
For such F, there exists y 2 DðlÞ such that ly ¼ Jy 0 �Hy ¼ WF . For all ½a; b� � I and g 2 D0ð½a; b�Þ, Z b
aðHgÞ�Wy ¼

Z b

ag�WF :

So we have provided supp y � [a,b] that
Z b
aðHgÞ�Wf ¼

Z b

aðHgÞ�Wy;

which implies f � y 2 R0ð½a;b�Þ?ð� L2W ð½a; b�ÞÞ. Let f � y = z for t 2 [a,b], we have lz ¼ Jz0 �Hz ¼ 0. Since

[a,b] is an arbitrary interval in I, we have f 2 DðlÞ and Hf ¼ Hy ¼ F 2 L2W ðIÞ. This means f 2 D0ðHÞ,

which implies DðH �0Þ � DðHÞ. The inverse part DðHÞ � DðH �0Þ is obvious. Moreover, H is a closed operator.


For all f 2 DðHÞ and g 2 DðhÞ, [f :g] = (Hf,g)W � (f,hg)W = 0, so we obtain f 2 Dðh�Þ and h*f = Hf. Thus,DðHÞ ¼ Dðh�Þ, and H = h*. By the definition of h, h is a closed-operator on L2

W ðIÞ. Since h** = h, we haveH* = h, H ��0 ¼ H 0. Thus, H 0 ¼ H ��0 ¼ H � ¼ h. The proof of Theorem 3.12 is complete. h

For the rest of the paper, let us make some standing hypotheses.By Theorem 3.12, the minimal operator h generated by l is densely defined, closed, symmetric on L2

W ðIÞ,and h is self-adjoint if and only if h = H. Therefore, in what follows, we always suppose h 5 H. If there existsa self-adjoint extension K = K* of h, then h � K � H, DðhÞ � DðKÞ � DðHÞ. Furthermore, the domain of self-adjoint operator K can be obtained by imposing suitable boundary conditions on functions in DðHÞ (seeTheorem 4.5 below).

The existence of self-adjoint extension of minimal operator h is determined by the structure of linear spaceDðHÞ. Since H is a closed operator, DðHÞ is a Hilbert space equipped with the graph-norm

ðx; yÞ� ¼ ðx; yÞW þ ðHx;HyÞW ; 8x; y 2 DðHÞ
(see [11]), and the boundary form [:] is continuous. DðhÞ has a direct sum decomposition in DðHÞ:
DðHÞ ¼ DðhÞ �D� �Dþ;

where D� and Dþ are the deficiency spaces of DðhÞ in L2W ðIÞ,

D� ¼ spanff 2 Dðh�Þjh�f ¼ �if g:

d� ¼ dimD� are called the deficiency indices of DðhÞ in L2W ðIÞ, and 0 6 d±

6 2n. There exists self-adjointextension of h in L2

W ðIÞ if and only if d� = d+ (see [8, chapter IV]). In what follows, we always suppose

d ¼ d� ¼ dþ: ð10Þ

4. GKN theory for linear Hamiltonian system

In this section, we establish the GKN theory for linear Hamiltonian system, namely, we obtain the self-adjoint extension of the minimal operator h by imposing suitable boundary conditions on the maximal oper-ator H and examine the geometric and algebraic structure of the set of all self-adjoint extensions of h.

Definition 4.1. DðhÞ is a linear subspace of complex Hilbert space DðHÞ (in the sense of graph-norm), wedefine the quotient space

Ł ¼ DðHÞ=DðhÞ ¼ fff þDðhÞgjf 2 DðHÞg
as an endpoint space.
The endpoint space Ł is a complex vector space with dimension 2d, we denote the quotient projection by

W : DðHÞ ! Ł

f 7! Wf ¼ ff þDðhÞg:

For convenience, we denote Wf ¼ f for given f 2 DðHÞ, i.e., f 2 Ł is a coset. If there exists u 2 DðHÞ suchthat [f :u] = 0 for all f 2 DðHÞ, then we call [f :u] = 0 is a boundary condition. Boundary condition is a linearfunctional in DðHÞ, and it is continuous in the sense of symplectic form. Suppose [f :u1] = 0,[f :u2] = 0, . . . , [f :um] = 0 are boundary conditions, if the unique solutions of

Pmr¼1cr½f : ur� ¼ 0 for all

f 2 DðHÞ are c1 = c2 = � � � = cm = 0, then we call the boundary conditions are linear independent. By the def-inition, the boundary conditions [f :u1] = 0, [f :u2] = 0, . . . , [f :um] = 0 are linear independent if and only if½f : u1� ¼ 0; ½f : u2� ¼ 0; . . . ; ½f : um� ¼ 0 are linear independent. The linear independent boundary conditions[f :u1] = 0, [f :u2] = 0, . . . , [f :um] = 0 are called self-adjoint, if [ur :uk] = 0 for all r, k = 1,2, . . . ,m. Two groupsof boundary conditions [f :ur] = 0 and [f :vr] = 0 (r = 1,2, . . . ,m) are called equivalent, if f 2 DðHÞ satisfiesone group of equations, then f satisfies the another and vice versa.

Definition 4.2. The collection ff r j r ¼ 1; 2; . . . ; dg � L2W ðIÞ is called a boundary condition set or GKN set of

operator pairs {h,H}, if


1. f r 2 DðHÞ; r ¼ 1; 2; . . . ; d.2. {[f : f r] = 0jr = 1,2, . . . ,d} are linear independent boundary conditions.3. {[f : f r] = 0jr = 1,2, . . . ,d} are self-adjoint.

Remark 4.3. {f r j r = 1,2, . . . ,d} is a GKN set of operator pairs {h,H} if and only if f bf r j r ¼ 1; 2; . . . ; dg is aLagrangian d-subspace of Ł. Since Ł is a 2d-dimensional complex vector space, so f bf r j r ¼ 1; 2; . . . ; dg is acompleted Lagrangian subspace of Ł.

Lemma 4.4. ðDðHÞ; ½:�Þ is a pre-symplectic space, DðhÞ is a Lagrangian subspace. The endpoint space Ł, with the

symplectic structure inherited by DðHÞ, i.e.,

½f : g� ¼ ½f : g� for all f ; g 2 DðHÞ

is a symplectic space.

Theorem 4.5. Suppose system (2) is controllable and (10) holds, H and h are defined as above. Then there exists a

natural one-to-one corresponding between the set {K} of self-adjoint operator K of h on DðKÞ � DðHÞ and the set

{L} of all Lagrangian d-space L in the complex symplectic 2d-space Ł. Namely, suppose {f r j r = 1,2, . . . , d} is a

GKN set of operator pairs {h,H}, let

D ¼ f 2 DðHÞ j ½f : f r� ¼ 0; r ¼ 1; . . . ; dn o

¼ c1f 1 þ c2f 2 þ � � � þ cdf d þDðhÞ; 8c1; c2; . . . ; cd 2 C;

ð11Þ
we define an operator as follows:
K : DðKÞ ¼ D! L2W ðIÞ; Kf ¼ Hf :

Then K is a self-adjoint extension of h (K is an operator confining H with boundary conditions

[f : f r] = 0, r = 1,2, . . . , d).

Conversely, if K is a self-adjoint extension of h, there exist f 1; f 2; . . . ; f d 2 DðHÞ, such that the domain of K

can be written as (11), and {fr j r = 1,2, . . . , d} is a GKN set of operator pairs {h,H}.

Remark 4.6. If d = 0, then h = H is the unique self-adjoint operator, and Ł = {0}, Theorem 4.5 holds obvi-ously. Now, suppose 0 < d 6 2n, so there exists verily self-adjoint extension K of h, satisfyingDðhÞ � DðKÞ � DðHÞ. Since the self-adjoint extension K of operator h is determined by DðKÞ, we necessarilyprove that there exists a one-to-one corresponding relationship between fDg ¼ fDðKÞg and{L} = {GKN set}.

Firstly, we give the following lemmas, which will be useful in the proof of Theorem 4.5, and they are ofindependent interest in their own right.

Lemma 4.7. A linear manifold D in L2W ðIÞ is the domain of certain self-adjoint extension of operator h if and only

if

(a) DðhÞ � D � DðHÞ;(b) ½f : g� ¼ 0; 8f ; g 2 D;(c) f 2 DðHÞ; 8g 2 D, [f :g] = 0 implies f 2 D.

Proof. Necessity. Suppose K is a self-adjoint extension of h, then DðhÞ � D ¼ DðKÞ. Since h � K and K isself-adjoint, K = K* � h* = H, and DðKÞ � DðHÞ. Thus, we obtain (a). Since 8f ; g 2 DðKÞ, K is a restrictionof H,

½f : g� ¼ ðHf ; gÞW � ðf ;HgÞW ¼ ðKf ; gÞW � ðf ;KgÞW ¼ 0;


which implies (b) holds. f 2 DðHÞ, 8g 2 D, [f :g] = 0, i.e., (Hf,g)W � (f,Kg)W = 0, 8g 2 DðKÞ. Sof 2 DðK�Þ ¼ DðKÞ, and (c) is true.

Sufficiency. Define an operator K as follows:

K : D! L2W ðIÞ; Kf ¼ Hf :

(1) The domain of K is dense. This follows from (a) and Theorem 3.9.(2) K is Hermitian. By (b), 8f ; g 2 D, (Kf,g) = (f,Kg). Moreover, we have DðKÞ � DðK�Þ.(3) K is self-adjoint. For each f 2 DðK�Þ, by the definition of conjugate operator, 8g 2 DðKÞ,

(f,Kg) = (Hf,g), by (c), we have f 2 D, so DðK�Þ � DðKÞ. This means DðK�Þ ¼ DðKÞ. So K is a self-adjoint operator generated by operator H. h

Since d+ = d�, there exists a unitary transformation between Dþ and D�

U : Dþ ! D�: ð12Þ
The following lemma gives the relationship between unitary transformation and the self-adjoint extension ofoperator h.
Lemma 4.8. The linear manifold D � L2W ðIÞ is a domain of self-adjoint extension of operator h if and only if

there exists a unitary transformation U : Dþ ! D�, such that

D ¼ f 2 DðHÞ j f ¼ hþ ðI � UÞ/; h 2 DðhÞ;/ 2 Dþf g: ð13Þ

Proof. Necessity. Suppose K is a self-adjoint extension of h, Define the Cayley transformationU = (K � i)(K + i)�1: R(K + i)! R(K � i). Since K is self-adjoint, RðK � iÞ ¼ L2

W ðIÞ, and U is a unitarytransformation from Dþ to D�, we define U0 = (h � i)(h + i)�1, then K = i(I + U)(I � U)�1,h = i(I + U0)(I � U0)�1. Now,

DðKÞ ¼ f 2 DðHÞ j f ¼ ðI � UÞg; g 2 L2W ðIÞ

� �;

DðhÞ ¼ f 2 DðHÞ j f ¼ ðI � U 0Þg; g 2 L2W ðIÞ

� �:

Since

L2W ðIÞ ¼ Rðhþ iÞ � ½Rðhþ iÞ�? ¼ Rðhþ iÞ � NðH � iÞ

for all f 2 DðKÞ, there exists g 2 L2W ðIÞ such that f = (I � U)g. Suppose g = g1 + g2, where g1 2 R(h + i),

g2 2 N(H � i), then

f ¼ ðI � UÞðg1 þ g2Þ ¼ ðI � UÞg1 þ g2 � Ug2;

where ðI � UÞg1 2 DðhÞ, g2 2 N(H � i), which mean g2 2 Dþ, Ug2 2 D�. So the elements of DðKÞ can be writ-ten as (13).

Sufficiency. It is sufficient to prove that the operator K, with domain (13), is self-adjoint. By Lemma 4.7, it issufficient to verify D satisfying (a)–(c). (a) holds obviously. For all f ; g 2 D, suppose f = h1 + /1 + U/1,g = h2 + /2 + U/2, where /1;/2 2 Dþ, by the definition of boundary form,

½f : g� ¼ ½h1 þ /1 þ U/1 : h2 þ /2 þ U/2� ¼ ½/1 : /2� þ ½U/1 : U/2� þ ½U/1 : /2� þ ½/1 : U/2�:
Since [/1 :/2] = 2i(/1,/2)W, [U/1 :U/2] = �2i(U/1,U/2)W = �2i(/1,/2)W,[U/1 :/2] = (/1,U/2)W = 0, so[f :g] = 0. Consequently, (b) is true. Suppose f 2 DðHÞ, without loss of generality, we may supposef = h1 + f1 + f2, h1 2 DðhÞ, f1 2 Dþ, f2 2 D�. If for all g 2 D, [f :g] = 0 implies f 2 D. Supposeg = h2 + g1 + Ug1, we have 0 = [f :g] = 2i{(f1,g1)W � (f2,Ug1)W}, i.e., (f1,g1)W = (f2,Ug1)W. Since U conservesorthogonality and g1 2 Dþ is arbitrary, we get f2 = Uf1, i.e., (c) holds. Now the conclusion holds. h
Since H is a closed operator, Dþ;D� are close d-subspaces of L2W ðIÞ. Assume {/r j r = 1,2, . . . ,d} are nor-

malized conjoined bases of Dþ, and {wr j r = 1,2, . . . ,d} are normalized conjoined bases of D�, for given/ 2 Dþ, suppose / ¼

Pdr¼1ar/r, ar 2 Cð1 6 r 6 dÞ are constants. Then


U/ ¼Xd

r¼1

arU/r ¼Xd

r¼1

ar

Xd

k¼1

urkwk;

where U ¼ ðurkÞd�d is a unitary matrix. By Lemma 4.8, we have the following corollary.

Corollary 4.9. Suppose K is a self-adjoint extension of h, its domain is (13). If {/r j r = 1,2, . . . , d} are the

normalized conjoined bases of Dþ, {wrjr = 1,2, . . . , d} are the normalized conjoined bases of D�, then there exists

unitary matrix U = (urk)d·d such that DðKÞ can be written as follows:

DðKÞ ¼ f 2 DðHÞ j f ¼ hþXd

r¼1

arWr; h 2 DðhÞ; ar 2 C;Wr ¼ /r �Xd

k¼1

urkwr

( ): ð14Þ

Lemma 4.10. Suppose K is a self-adjoint extension of h, its domain is determined by (14). Then DðKÞ can be writ-

ten as follows:

DðKÞ ¼ f 2 DðHÞ j ½f : Wr� ¼ 0; r ¼ 1; 2; . . . ; df g: ð15Þ

Proof. It is sufficient to prove D ¼ DðKÞ. By the definition of Wr and Green formula, Wr 2 DðKÞ. For allu 2 DðKÞ, suppose u ¼ hþ

Pdr¼1arWr, h 2 DðhÞ, a 2 C, 1 6 r 6 d, then we get by Green formula,

[/r :/k] = 2idrk, [wr :wk] = � 2idrk, [/r :wk] = 0, (r,k = 1,2, . . .d). Consequently,

½Wr : Wk� ¼ ½/r : /k� �Xd

s¼1

urs½wr : /k� �Xd

l¼1

ukl½/r : wl� þXd

s;l¼1

ursukl½ws : wl� ¼ 2idrk � 2iXd

s¼1

ursuks ¼ 0:

ð16Þ
Consequently,
½u : Wk� ¼ ½h : Wk� þXd

r¼1

arWr : Wk

" #¼Xd

r¼1

ar½Wr : Wk� ¼ 0;

which implies u 2 D, DðKÞ � D. Conversely, since [u :Wk] = 0, 1 6 k 6 d, suppose u = h + u+ + u�, whereh 2 DðhÞ, uþ 2 Dþ, u� 2 D�, then there exist br, cr, such that

u ¼ hþXd

r¼1

br/r þXd

r¼1

crwr:

Since [u :Wk] = 0, we have br ¼ �Pd

k¼1ckurk. U�U ¼ I , we obtain cr ¼ �Pd

k¼1bkukr. Now,

u ¼ hþXd

r¼1

br/r �Xd

r¼1

Xd

k¼1

bkukrwr ¼ hþXd

r¼1

br /r �Xd

k¼1

urkwk

!¼ hþ

Xd

r¼1

brWr:

So u 2 DðKÞ. Consequently, D � DðKÞ. This completes the proof. h

Lemma 4.11. Suppose K is a self-adjoint extension of h, Wr is defined as in (14). Then we obtain

(a) [Wr :Ws] = 0,1 6 r, s 6 d,

(b) {Wr j r = 1,2, . . . , d} is a GKN set of operator pairs {h, H}.

Proof. Obviously, (a) is true by (16). Now we prove (b) is true. Firstly, Wr 2 Dþ �D� � DðHÞ. Secondly,suppose
Xd
r¼1

cr½f : Wr� ¼ f :Xd

r¼1

crWr

" #¼ 0:


By the definition of DðhÞ, we obtainPd

r¼1crWr 2 DðhÞ, i.e.,
Xd
r¼1

cr/r �Xd

r¼1

cr

Xd

k¼1

urkwr ¼ 0:

Since Dþ \D� ¼ ;,Pd

r¼1cr/r ¼Pd

r¼1crPd

k¼1urkwr ¼ 0, we have c1 = c2 = � � � = cd = 0, i.e., {[f :Wr] = 0 j r =1,2, . . . ,d} are linearly independent boundary conditions. By (16), [Wr :Wk] = 0. So [f :Wr] = 0(1 6 r 6 d) is aself-adjoint boundary condition set. Thus, {Wr j r = 1,2, . . . ,d} is a GKN set of operator pairs {h,H}. h

We are now in a position to proceed with the proof of Theorem 4.5.

Proof of Theorem 4.5. Firstly, suppose K is a self-adjoint extension of h, by Lemma 4.8 and Corollary 4.9,there exists a unitary matrix U ¼ ðurkÞd�d such that (13) and (14) are true. Define fr = Wr, r = 1,2, . . .d, thenwe have {Wr j r = 1,2, . . . ,d} is a GKN set of operator pairs {h,H} by Lemma 4.11.

Now, we turn to prove the first part. Suppose {f1, f2, . . . , fd} is a GKN set of operator pairs {h,H}, we definelinear manifold D � DðHÞ as follows:

D ¼ ff 2 DðHÞ j ½f : f r� ¼ 0; r ¼ 1; 2; . . . ; dg:
Define
D ¼ ff 2L j ½f : bf r � ¼ 0; r ¼ 1; 2; . . . ; dg:
Then f 2 D () f 2 bD. Let D0 ¼ spanf bf1 ; bf2 ; . . . ; bfdg, we have D0 � bD. Now, we prove D0 ¼ bD. SincedimD0 ¼ d, we get dim bD P d. Moreover, D is a Lagrangian subspace of Ł and dim Ł = 2d, so dim bD ¼ d,so we obtain D0 ¼ bD. Now we define operator as follows:
K : D! L2W ðIÞ; Kf ¼ Hf :

Next, we prove D satisfying Lemma 4.7(a)–(c). For all f 2 DðhÞ, g 2 DðHÞ, by the definition of DðhÞ,[f :g] = 0. Particularly, [f : fr] = 0, r = 1,2, . . . ,d, so f 2 D, and D � DðHÞ is obvious, which shows (a) is true.For all f ; g 2 D, suppose g ¼ hþ

Pdr¼1arf r, then

½f : g� ¼ ½f : h� þ f :Xd

r¼1

arf r

" #¼Xd

r¼1

ar½f : f r�:

[f : f r] = 0 implies [f :g] = 0, so (b) is satisfied. For all f 2 DðHÞ, g 2 D, [f :g] = 0. Since g is selected arbitrarily,let g = f r, we obtain [f : f r] = 0, r = 1,2, . . . ,d, which means f 2 D. Condition (c) is true. The proof is completenow. h

Remark 4.12. By the proofs of Lemma 4.7 and Theorem 4.5, we can deduce the following facts: K is a self-adjoint extension of h if and only if there exists a unitary transformation U : Dþ ! D�, such that DðKÞ can bewritten as (13) or (14). In this case, L = graphU.

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gkn theory for linear hamiltonian systems

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