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arXiv:1907.12218 · Progress of Theoretical and Experimental Physics

Exactly solvable discrete quantum mechanical systems and multi-indexed orthogonal polynomials of the continuous Hahn and Meixner–Pollaczek types

Satoru Odake

Published Oct 11, 2019• 8 citations
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Abstract

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 $\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}$, where $\ell_{\mathcal{D}}$ is an even positive integer ($\mathcal{D}$: a multi-index set), but they form a complete set of orthogonal basis in the weighted Hilbert space.

DOI
10.1093/ptep/ptz124

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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.
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.
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...........
Abstract We present a formulation of quantum mechanics based on the theory of orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis where the expansion coefficients are orthogonal polynomials in the energy and physical parameters. Information about the corresponding physical systems (both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable quantum mechanical problems becomes larger than in the conventional formulation (see, for example, table 3 in the text). We limit our investigation in this work to the Askey classification scheme of hypergeometric orthogonal polynomials and focus on the Wilson polynomial and two of its limiting cases (the Meixner–Pollaczek and continuous dual Hahn polynomials). Nonetheless, the formulation is amenable to other classes of orthogonal polynomials.
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.
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.