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arXiv:1702.03078 · Progress of Theoretical and Experimental Physics

New determinant expressions of multi-indexed orthogonal polynomials in discrete quantum mechanics

Satoru Odake

Published Apr 4, 2017• 4 citations
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Abstract

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...........

DOI
10.1093/ptep/ptx051

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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...........
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.
Abstract We present the case-(1) multi-indexed orthogonal polynomials of a discrete variable for 8 types ((dual)(qqq-)Hahn and three kinds of qqq-Krawtchouk and qqq-Meixner). Based on them and the case-(1) multi-indexed orthogonal polynomials of Racah, qqq-Racah, Meixner, little qqq-Jacobi, and little qqq-Laguerre types, exactly solvable continuous time birth and death processes are obtained. Their discrete time versions (Markov chains) are also obtained for finite types.
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.
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.
Abstract We present new exactly solvable systems of the discrete quantum mechanics with pure imaginary shifts, whose physical range of coordinates is a whole real line. These systems are shape invariant and their eigenfunctions are described by the multi-indexed continuous Hahn and Meixner–Pollaczek orthogonal polynomials. The set of degrees of these multi-indexed polynomials is {ℓD,ℓD+1,ℓD+2,…}\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}{ℓD​,ℓD​+1,ℓD​+2,…}, where ℓD\ell_{\mathcal{D}}ℓD​ is an even positive integer (D\mathcal{D}D: a multi-index set), but they form a complete set of orthogonal basis in the weighted Hilbert space.