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arXiv:1708.01830 · Progress of Theoretical and Experimental Physics

Casoratian identities for the discrete orthogonal polynomials in discrete quantum mechanics with real shifts

Satoru Odake

Published Oct 31, 2017• 2 citations
arXivHTMLPDF

Abstract

In our previous papers [S. Odake and R. Sasaki, J. Phys. A 46, 245201 (2013) and S. Odake and R. Sasaki, J. Approx. Theory 193, 184 (2015)], the Wronskian identities for the Hermite, Laguerre, and Jacobi polynomials and the Casoratian identities for the Askey–Wilson polynomial and its reduced-form polynomials were presented. These identities are naturally derived through quantum-mechanical formulation of the classical orthogonal polynomials: ordinary quantum mechanics for the former and discrete quantum mechanics with pure imaginary shifts for the latter. In this paper we present the corresponding identities for the discrete quantum mechanics with real shifts. Infinitely many Casoratian identities for the |$q$|-Racah polynomial and its reduced-form polynomials are obtained.

DOI
10.1093/ptep/ptx165

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In our previous papers [S. Odake and R. Sasaki, J. Phys. A 46, 245201 (2013) and S. Odake and R. Sasaki, J. Approx. Theory 193, 184 (2015)], the Wronskian identities for the Hermite, Laguerre, and Jacobi polynomials and the Casoratian identities for the Askey–Wilson polynomial and its reduced-form polynomials were presented. These identities are naturally derived through quantum-mechanical formulation of the classical orthogonal polynomials: ordinary quantum mechanics for the former and discrete quantum mechanics with pure imaginary shifts for the latter. In this paper we present the corresponding identities for the discrete quantum mechanics with real shifts. Infinitely many Casoratian identities for the |qqq|-Racah polynomial and its reduced-form polynomials are obtained.
Multi-indexed orthogonal polynomials (the Meixner, little q-Jacobi (Laguerre), (q-)Racah, Wilson, and Askey-Wilson types) satisfying second-order difference equations were constructed in discrete quantum mechanics.They are polynomials in sinusoidal coordinates η(x) (x is the coordinate of the quantum system) and are expressed in terms of Casorati determinants whose matrix elements are functions of x at various points.By using shape-invariance properties, we derive various equivalent determinant expressions, especially those whose matrix elements are functions of the same point x.Except for the (q-)Racah case, they can be expressed in terms of η only, without explicit x-dependence...........
A quantum orthogonal polynomial transform (QOPT) is an algorithmic primitive that maps a superposition of standard basis states ∑kαk∣k⟩\sum_k α_{k} \ket{k}∑k​αk​∣k⟩ coherently to a basis of normalized univariate polynomials orthogonal with respect to a probability measure μ(x)μ(x)μ(x). We provide new discrete and continuous QOPTs for polynomial families in the Askey scheme extending the results for the quantum Hermite transform (Jain et al., STOC'26). Our efficient QOPT algorithms require time O(polylog(N,1/ε))O(\text{polylog}(N, 1/ε))O(polylog(N,1/ε)), where NNN is the grid size and εεε is the error, for the discrete Charlier, Meixner and Krawtchouk transforms and for the integer-order Laguerre transform in the continuous setting. The efficient QOPTs are obtained by uncovering the links between Askey scheme polynomials and Gaussian quantum optical gates and developing a compilation framework for SU(2)SU(2)SU(2) and SU(1,1)SU(1,1)SU(1,1) optical gates on Cartesian grids extending the framework developed by Iyer et al. (arXiv:2602.15180). Further, we reduce the continuous Jacobi transform to the discrete Hahn transform and provide an O ⁣(Npolylog⁡((N+α+β+1)/ε))O\!\left(N\operatorname{polylog}((N+α+β+1)/ε)\right)O(Npolylog((N+α+β+1)/ε))-time Hahn transform, a quadratic speedup over the naive implementation. This is based on a more efficient compilation of the Clebsch--Gordan transform for coupling SU(2)\mathrm{SU}(2)SU(2) representations with spins (j1,j2)(j_{1}, j_{2})(j1​,j2​). Finally, we develop a new framework for Laguerre transforms for all orders ν>0ν>0ν>0 by fast-forwarding the corresponding radial oscillator using a 3-term chirp decomposition and an efficient algorithm for the Quantum Hankel Transform on a logarithmic grid.
Multi-indexed orthogonal polynomials (the Meixner, little q-Jacobi (Laguerre), (q-) Racah, Wilson, and Askey-Wilson types) satisfying second-order difference equations were constructed in discrete quantum mechanics. They are polynomials in sinusoidal coordinates eta(x) (x is the coordinate of the quantum system) and are expressed in terms of Casorati determinants whose matrix elements are functions of x at various points. By using shape-invariance properties, we derive various equivalent determinant expressions, especially those whose matrix elements are functions of the same point x. Except for the (q-) Racah case, they can be expressed in terms of eta only, without explicit x-dependence.
We consider dual polynomials of the multi-indexed (⁠|qqq|-)Racah orthogonal polynomials. The |MMM|-indexed (⁠|qqq|-)Racah polynomials satisfy the second-order difference equations and various |1+2L1+2L1+2L| (⁠|L≥M+1L\geq M+1L≥M+1|⁠) term recurrence relations with constant coefficients. Therefore their dual polynomials satisfy the three term recurrence relations and various |2L2L2L|th-order difference equations. This means that the dual multi-indexed (⁠|qqq|-)Racah polynomials are ordinary orthogonal polynomials of Krall type. We obtain new exactly solvable discrete quantum mechanics with real shifts, whose eigenvectors are described by the dual multi-indexed (⁠|qqq|-)Racah polynomials. These quantum systems satisfy the closure relations, from which the creation/annihilation operators are obtained, but they are not shape invariant.
Various examples of exactly solvable ‘discrete’ quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilation-creation operators, the dynamical symmetry algebras and coherent states. The eigenfunctions are the (q-) Askey-scheme of hypergeometric orthogonal polynomials satisfying difference equation versions of the Schrödinger equation. Various reductions (restrictions) of the symmetry algebra of the Askey-Wilson system are explored in detail.
It is a well established fact that there are deeper connections between the theory of orthogonal polynomials on the one hand and properties of Schrodinger operators on the other hand. Those operators are assumed to act in conventionally used Hilbert spaces like for example L 2 (R n ). A pro- minent example for these connections are the classical continuous Hermite polynomials which correspond to Schrodinger operators with a quadratic po- tential. In the one dimensional case, the support of these polynomials is the real line. When dealing with a discretization of this support, one meets the next ingredient which enriches the investigation of orthogonal polynomials: It is the aspect of deformation. The idea of deforming polynomials plays a crucial role in the context of special functions. In the case of a q deformation one sees that the deformation itself can be associated with discretizing the support for orthogonal polynomials. This allows to switch from polynomials dened in the continuum to polynomials dened on a geometric progression. Behind these observations seems to lie a more general concept: Deformati- ons can be related to discretizations or quantizations and vice versa. This concept has also turned out to be an important guideline in structures of non-commutative geometry, thus also in areas quite far away from the theory of orthogonal polynomials. Our purpose is to address to generalized Hermite polynomials on the lattice- like support f+q n ; q n jn 2 Zg. These polynomials will turn out to be one- parameter generalizations of the q discrete Hermite polynomials of type II. Our basic starting point is the introduction of dierence operators which transform these polynomials into each other. These operators contain two parameters, the deformation parameter q which is related to the discrete support f+q n ; q n jn 2 Zg and a parameter which couples one-dimensional Dunkl like operators Mq ;M + q to the dierence operators under consideration. The parameter can be regarded as a further deformation parameter. Having derived the q generalized Hermite polynomials in the course of the second chapter, we classify them. It turns out that they are related to the discrete monic q Laguerre polynomials L () n (x; q). We investigate the related dierence operators in more detail and establish a link to the moment problem for the q generalized Hermite polynomials.