Satoru Odake (Shinshu University)

Collaboration with Ryu Sasaki (YITP)

§1. Introduction §2. Exceptional orthogonal polynomials and infinitely many shape invariant quantum mechanical systems §3. Summary and comments

title

(Exceptional) Orthogonal Polynomials and (Discrete) Quantum Mechanics

S.O. and Ryu Sasaki, arXiv:0902.2593, 0906.0142, 0909.3668, 0911.1585, 0911.3442, 1004.0544, 1007.3800, 1101.5468, 1102,0812, 1104.0473, 1105.0508.

C.-L.Ho, S.O. and Ryu Sasaki, arXiv:0912.5447. L.Garcia-Gutierrez, S.O. and Ryu Sasaki, arXiv:1004.0289.

2011.5.9 NU string seminars at Nagoya University

Multi-indexed Orthogonal Polynomials arXiv:1105.0508

intro1

Schrödinger eq.

§1. Introduction

eigenfunctions

1dim.

ordinary QM

2nd order differential eq.

Sturm-Liouville problem

岩波数学辞典（第４版） １８６ 常微分方程式の境界値問題 １８６C．Sturm-Liouville の問題

intro1a

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intro2

Sturm-Liouville’s theorem

discrete eigenvalues

non-degenerate

：n nodes, orthogonal functions choose constant term

Existence theorem. Whether and are expressed explicitly is another problem.

factorization

intro3

energy eigenvalues and eigenfunctions are obtained explicitly.

Exactly Solvable

Shape invariance

sufficient condition for ES

parameter(s):

in detail

Intro4

case

: orthogonal polynomial

weight function:

： sinusoidal coordinate exact Heisenberg solution annihilation/creation operators

Ex: exactly solvable systems（shape invariant）

harmonic oscillator

Hermite polynomial

radial oscillator

Laguerre polynomial

Darboux-Pöschl-Teller pot.

Jacobi polynomial

Intro5

differential equation

Hermite polynomial

Laguerre polynomial

Jacobi polynomial

intro6

Bochner’s theorem (1929)

（modulo affine transformation of x）

: Hermite, Laguerre, Jacobi, Bessel polynomial

：orthogonal poly. （degree n）

satisfy 2nd order differential equation

To avoid this no-go theorem・・・

2nd order higher order

differential eq. difference eq.

exceptional orthogonal polynomial

‘discrete’quantum mechanics

・

・

・

（x ：continuous, discrete）

weight function is not positive definite

2-1

§２. Exceptional orthogonal polynomials and infinitely many shape invariant quantum mechanical systems

（ordinary）orthogonal polynomial

complete

orthonormal system weight function given

Gram-Schmidt’s method orthogonal polynomial

X1-Laguerre, X1-Jacobi

・Gomez-Ullate - Kamran - Milson arXiv:0807.3939

‘good’basis starting at degree 1 and weight function orthonormalization

weight function

satisfy 2nd order differential equation

2-2

weight function

basis

satisfy 2nd order differential eq.

weight function

basis

degree n +1

satisfy 2nd order differential eq.

degree n +1

2-3

・Quesne arXiv:0807.4087

Construction of quantum mechanical systems whose eigenfunctions are described by X1- Laguerre or X1- Jacobi polynomials. They are shape invariant.

2-4

Infinitely many shape invariant systems and Xℓ-orthogonal polynomials

deform radial osc. and DPT pot. keeping shape invariance

no nodes

well-defined

shape invariant

arXiv:0906.0142 0911.3442

Odake-Sasaki

2-5

eigenvalues and eigenfunctions

complete set

：exceptional orthogonal poly.

・ ・ ・

explicit form

more explicitly

・

2-5a

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2-6

Xℓ-Jacobi case Fuchsian differential equation

satisfies 2nd order differential equation

regular singularities and

the first examples of global solutions of Fuchsian differential equations having as many as 3+ℓ regular singularities

more explicitly

2-6a

back

3-1

We have constructed shape invariant quantum mechanical systems by modifying radial osc. and DPT pot. for each ℓ(ℓ=1,2,…) infinitely many ! Eigenfunctions are described by exceptional Laguerre, Jacobi orthogonal polynomials Pℓ,n (n=0,1,2,… ; degree ℓ+n) starting at degree ℓ

§３. Summary and Comments ・

・ ≠ system obtained by states deletion (Adler’s theorem)

・ possible to construct shape invariant systems with continuou ℓ. （Pℓ,n are no longer polynomials）

・

: constant

intertwiner

3-2

・ Discrete Quantum Mechanics

Schrödinger eq. : 2nd order difference equation

x : continuous case

shape invariance, Crum’s theorem, Adler’s theorem, etc are treated similarly.

For good choices of V(x) , the systems are shape invariant and eigenfunctions are described by continuous Hahn, Wilson, Askey-Wilson polynomials.

For each ℓ（ℓ=1,2,…）, the original systems can be deformed keeping shape invariance and exceptional continuous Hahn, Wilson, Askey-Wilson orthogonal polynomials are constructed.

Askey-scheme of (q-)hypergeometric orthogonal polynomials explicitly

explicitly

shift to imaginary direction

・

・

・

・

3-3

・ Discrete Quantum Mechanics

Schrödinger eq. : 2nd order difference equation

x : discrete case

shape invariance, Crum’s theorem, Adler’s theorem, etc are treated similarly.

For good choices of B(x),D(x), the systems are shape invariant and eigenfunctions are described by Racah, q-Racah,… polynomials.

For each ℓ（ℓ=1,2,…）, the original systems can be deformed keeping shape invariance and exceptional continuous Racah, q-Racah orthogonal polynomials are constructed.

Askey-scheme of (q-)hypergeometric orthogonal polynomials

shift to real direction

・

・

・

・

miop-1

Method of virtual states deletion

deletion of the lowest lying eigenstate ・

arXiv:1105.0508

Multi-indexed Orthogonal Polynomials

Crum

after M steps

ordinary QM

miop-2

deletion of M distinct eigenstates ・ Adler

miop-3

eigenstates :

・ Virtual state

virtual states :

boundary conditions

deletion of M distinct virtual states

miop-4

2 types of virtual states

miop-5

M+N virtual states deletion

miop-6

Process:

miop-7

multi-indexed orthogonal polynomials

multi-indexed orthogonal polynomials

denominator polynomial

eigenfunctions

miop-8

orthogonality relation

miop-9

shape invariance

forward and backward shift operators

miop-10

2nd order differential equation

relation to the exceptional orthogonal polynomials

miop-11

𝒅𝒋=0 cases

equivalence

The method of virtual states deletion is applicable to discrete quantum mechanics !

end

END

3-2-a1

Examples

OK

(1)

(2)

(3)

imaginary shift cases (continuous x)

・Hemiticity of (for in the next slide)

3-2-a2

(1)

(2)

(3)

・excited states

・spectrum

・ground state (1)

(2)

(3)

orthogonal polynomial of

3-2-a3

(1) continuous Hahn poly. deformation of Hermite poly. (2) Wilson poly. deformation of Laguerre poly. (3) Askey-Wilson poly. deformation of Jacobi poly.

(1)

(2)

(3)

(q-)Askey-scheme of hypergeometric orthogonal polynomials

・orthogonal polynomial

back

・shape invariance (1) (2) (3)

3-2-b1

Deformed system

deforming polynomial

3-2-b2

Eigenfunctions

Shape invariance

3-2-b3

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non-degenerate eigenvalue

crum-1

・Hamiltonian :

・Schrödinger equation :

‘oscillation theorem’

Crum’s theorem (ordinary) QM

1 degrees of freedom bound state

choose constant term

Sturm-Liouville problem

can be chosen real.

has n zeros.

can be chosen positive. : prepotential

describes the relationship between the original and the associated Hamiltonian systems, which are iso-spectral except for the lowest energy state. construction of a family of iso-spectral Hamiltonians

crum-2

・ground state :

・factorization :

(1955) Crum’s theorem

・first step :

original system

crum-3

・repeat this step :

iso-spectral Darboux transformation :

(except for the groundstate)

crum-4

crum-5

・expressions in terms of the determinant :

Wronskian

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