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arXiv:1805.00345 · Progress of Theoretical and Experimental Physics

Dual polynomials of the multi-indexed (q-)Racah orthogonal polynomials

Satoru Odake

Published Jun 8, 2018• 3 citations
arXivHTMLPDF

Abstract

We consider dual polynomials of the multi-indexed (⁠|$q$|-)Racah orthogonal polynomials. The |$M$|-indexed (⁠|$q$|-)Racah polynomials satisfy the second-order difference equations and various |$1+2L$| (⁠|$L\geq M+1$|⁠) term recurrence relations with constant coefficients. Therefore their dual polynomials satisfy the three term recurrence relations and various |$2L$|th-order difference equations. This means that the dual multi-indexed (⁠|$q$|-)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 (⁠|$q$|-)Racah polynomials. These quantum systems satisfy the closure relations, from which the creation/annihilation operators are obtained, but they are not shape invariant.

DOI
10.1093/ptep/pty076

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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 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.
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.
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.
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.
In [R. Oste and J. Van der Jeugt, arXiv: 1507.01821 [math-ph]] we classified all pairs of recurrence relations in which two (dual) Hahn polynomials with different parameters appear. Such pairs are referred to as (dual) Hahn doubles, and the same technique was then applied to obtain all Racah doubles. We now consider a special case concerning the doubles related to Racah polynomials. This gives rise to an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. Just as it was the case for (dual) Hahn doubles, the resulting two-diagonal matrix can be used to construct a finite oscillator model. We discuss some properties of this oscillator model, give its (discrete) position wavefunctions explicitly, and illustrate their behavior by means of some plots.