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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
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
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
・
2-6
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
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-3
eigenstates :
・ Virtual state
virtual states :
boundary conditions
deletion of M distinct virtual states
miop-7
multi-indexed orthogonal polynomials
multi-indexed orthogonal polynomials
denominator polynomial
eigenfunctions
miop-11
𝒅𝒋=0 cases
equivalence
The method of virtual states deletion is applicable to discrete quantum mechanics !
3-2-a1
Examples
OK
(1)
(2)
(3)
imaginary shift cases (continuous x)
・Hemiticity of (for in the next slide)
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)
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