Dual Polynomials of theMulti-Indexed (q-)RacahOrthogonal Polynomials
Satoru Odake (Shinshu University)
8min§1. Introduction§2. Multi-indexed Polynomials 8min
4min
§3. Dual Polynomials
4min§4. Exactly Solvable rdQM
§5. Summary and Comments 2min

15 November 2018
SIDE13 (Symmetries and Integrability of Difference Equations)
at Fukuoka (Japan), 11-17 November 2018
title

arXiv:1804.10352
1805.00345

§1. Introduction
Quantum Mechanics vs Orthogonal Polynomials・ Orthogonal polynomials often appear in quantum mechanical systems.
harmonic oscillator  Hermite polynomialhydrogen atom  Legendre polynomial (associated Legendre “polynomial”)
Laguerre polynomial
Pöschl-Teller potential  Jacobi polynomial

・ By using the known properties of orthogonal polynomials,
we can investigate quantum mechanical systems.
・ Conversely, by using quantum mechanical systems,
we can investigate unknown properties of new orthogonal polynomials.
 physicist’s approach to orthogonal polynomials
・ We consider quantum mechanical systems in one dimension.

eigenvalue problem of the Schrödinger equation
exactly solvable :1-1

・ (ordinary) orthogonal polynomial

 a complete set
orthogonal with respect to appropriate inner product three term recurrence relations
・ Bochner’s theorem(ordinary) orthogonal polynomials satisfying(ⅲ)

2nd order differential equation (with polynomial coefficients)
(ⅰ)

(ⅱ)

 Hermite, Laguerre, Jacobi, Bessel polynomials
・ To avoid this No-Go theorem(ⅰ) 2nd  higher  Krall polynomials(ⅱ) differential  difference

non positive definite inner product Askey-Wilson, q-Racah polynomials etc.
 generalizations of Bochner’s theorem

Askey-scheme of hypergeomtricorthogonal polynomials(ⅲ) ordinary  non ordinary ：in spite of missing degrees, a complete set
Gómez-Ullate–Kamran–Milson exceptional or multi-indexed polynomialsarXiv:0807.3939
1-2

(ⅰ) 2nd  higher (ⅱ) differential  difference (ⅲ) ordinary  non ordinary
・ Quantum Mechanical Systems
oQM : ordinary Quantum Mechanics
We consider three kindof systems (Hamiltonians). idQM : discrete QM with imaginary shifts
rdQM : discrete QM with real shiftswith R.Sasaki
QMsystemdynamicalvariable

Schrödingerequation

order

examples of orthogonal
polynomialsoQM

continuousdifferential eq.

2ndHermite, Laguerre, Jacobi

idQM

continuousdifference eq.

2ndMP, Wilson, Askey-Wilson etc.

rdQM

discretedifference eq.

2ndHahn, Racah, q-Racah etc.

(ⅰi) oQM  idQM, rdQM Askey-scheme of hypergeomtricorthogonal polynomials(ⅰii) deform oQM, idQM, rdQM systems by Darboux transformations
multi-indexed orthogonal polynomials(ⅰ) We have considered 2nd order so far, but can think of higher orders.
Today’s talk : dual (q-)Racah multi-indexed polynomials, which satisfy
higher order difference equations, and exactly solvable rdQM systems
1-3

・ Hamiltonian : hermitian operator
Forms of Hamiltonians (2nd order case)oQM

idQMrdQM・ parameters :If needed, we write λ-dependence :1-4

§2. Muti-Indexed Polynomials
Deformation of quantum mechanical systems
original: described by ordinary orthogonal
polynomial
type: label of seed solutions

exactly solvable system

Darbouxtransformation

degree of polynomial part
briefly

deformed: multi-index set

exactly solvable system: described by multi-indexed
orthogonal polynomial: expressed by denominator polynomial: expressed in terms ofdeterminant (Wronskian, Casoratian)choice ofseed solutions
2-1

type of seed solution

deformed system

virtual state

iso spectraleigenstate

state deletion

pseudo virtual state

state addition

・ eigenstate :multi-indexed polynomial : polynomial in: sinusoidal coordinate: weight function

・I : a set of missing degrees
case (1)case (2)The case (1) multi-indexed polynomials were constructed forLaguerre, Jacobi, (Askey-)Wilson, (q-)Racah cases.
O-Sasaki : arXiv:1105.0508, 1203.5868, 1207.5584

In the following we consider the case (1) polynomials.

2-2Multi-indexed polynomial constructed by quantum mechanicalformulation and Darboux transformationcase (1): label of system (Darboux transformation): label of polynomial n = number of zeros in the physical region(number of sign changing)
They are not ordinary orthogonal polynomials.
 They do not satisfy three term recurrence relations.
 They satisfy recurrence relations with more terms.

・ Recurrence relationsarXiv:1303.5820, 1410.8236, 1509.08213,1606.02836, 1804.10352(a) variable dependent
coefficients(b) constant coefficients(a)depends on M only ( structure of Darboux transformation)(b) various X ’s are possible.
2-3

・ Recurrence relations with constant coefficients

What X gives these relations?
Answer :various X ’s are possibleoQM (Laguerre, Jacobi)
idQM (Wilson, Askey-Wilson)rdQM (Racah, q-Racah)

2-4

arXiv:1804.10352

Miki-Tsujimoto : arXiv:1410.0183

§3. Dual Polynomials
Multi-indexed (q-)Racah polynomials  described by rdQM
：variable(coordinate)：label of polynomial

finite systemn = number of sign changing

・ orthogonality
normalized eigenvector:

・ 2nd order difference equation  Schrödinger equation for polynomial partrecurrence relations for x
・ recurrence relations with constant coefficientsrecurrence relations for n3-1

orthogonality:

Dual multi-indexed (q-)Racah polynomialsarXiv:1805.00345

dual polynomial : exchange x and n・ orthogonalitynd order difference2
three term recurrence recurrence relations
・for label xequation ofrelations of ordinary orthogonal polynomialrecurrence relations

for variable x

・ 2L+1 term recurrence  2L-th order difference
relations ofequation ofrecurrence relations
 Krall typefor label n・ correspondence
3-2

recurrence relations
for variable n§4. Exactly Solvable rdQM
discrete QM with real shifts rdQM
dynamical variable
Hamiltonian:

finite system: hermitian matrix (real symmetric matrix)

2nd order casetri-diagonal matrix (Jacobi matrix)

at boundaries

・ Schrödinger equation
eigenvector

diagonalization of matrices
4-1(trivial for finite systems)

new exactly solvable rdQM systems

arXiv:1805.00345

For each: real symmetric(2L+1) - diagonal matrix

・ eigenvectors

Eigenvectors do not depend on X !
 For X1, X2 with arbitrary Y1, Y2
・closure relation holds (in brute force way)  creation/annihilation operators
shape invariance does not hold4-2

 first example

§5. Summary and Comments
・ By using QM systems, we study orthogonal polynomials.
relations for case (1) multi-indexed polynomials
・ Recurrence(variable dependent coefficients and constant coefficients).
(known for Laguerre, Jacobi, Wilson, Askey-Wilson)(q-)Racah arXiv:1804.10352

Dual multi-indexed (q-)Racah polynomials. arXiv:1805.00345
・ ordinary orthogonal polynomials,
higher order difference equations (Krall type).

・ New exactly solvable rdQM systems.

arXiv:1805.00345

eigenvectors : dual multi-indexed (q-)Racah polynomials

・ Future problems :・explicit form offor general n,k,X,D ?
・relation amongfor different X ?
・Is it possible to deform rdQM systems of dual multi-indexed(q-)Racah polynomials by Darboux transformation ?
・Is it possible to construct “dual polynomials’’ of multi-indexed(Askey-)Wilson polynomials ?
5-1

Ortho. Poly. and QM systems
How to avoid Bochner’s theorem
dyn. Schrödinger
QM
equation
system var.

(ⅱ)

2nd
differential
equationdifference
equation(ⅱ)

difference
equation(ⅱ)

examples of orthogonal
polynomials

status

Hermite, Laguerre, Jacobi◎

multi-indexed version (ⅲ)

〇higher Krall version (i)(ⅰ)
multi-indexed version (i)(iii)
2nd◎

multi-indexed version (ii)(ⅲ)△△×Hahn, Racah, q-Racah etc.

◎multi-indexed version (ii)(ⅲ)△ higher Krall type version (i)(ii)(ⅰ)
multi-indexed version (i)(ii)(iii)
understanding : ◎＞〇＞△＞×

5-2

×MP, Wilson, Askey-Wilson etc.

higher Krall type version (i)(ii)(ⅰ)
multi-indexed version (i)(ii)(iii)
2nd〇

△×

today’s talk(ⅱ)

discreterdQM

continuous

idQM

continuousoQM

order(ⅰ) 2nd order  higher(ⅱ) differential  difference(ⅲ) ordinary  non-ordinary

END