Discrete reflectionless potentials, quantum algebras, and \(q\)-orthogonal polynomials
type: Item
instance of: scholarly article
title: Discrete reflectionless potentials, quantum algebras, and \(q\)-orthogonal polynomials
zbMATH Open document ID: 0813.35098
author: Alexei Zhedanov
author: V. P. Spiridonov
published in: Annals of Physics
publication date: 1995-03-02T00:00:00Z
review text: Using Darboux transformations for the lattice Schrödinger equation, we construct two families of discrete reflectionless potentials with \(j\) and \(2j\) bound states (solitons). These potentials are related to the exceptional Askey-Wilson polynomials. In the continuous limit they are reduced to the well-known solvable potential \(u(x) = - j(j+1)/ \cosh^ 2 x\). The limit \(j \to \infty\) for the lattice systems may be defined in two different ways. In the first case, discrete spectrum grows exponentially fast, and the continuous spectrum band, \(-2 \leq \lambda \leq 2\), is preserved. In the second case, the limiting potentials are related to the \(q^{-1}\)-Hermite polynomials. Their spectra are purely discrete and consist of one or two geometric series accumulating near the zero. These series are generated by the \(q\)-Weyl algebra and its ``square root'' version. In contrast to the continuous reflectionless potentials with quantum algebraic symmetries, the derived discrete potentials are expressed in terms of the elementary functions.
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 33D55
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 82B20
zbMATH DE Number: 727781
zbMATH Keywords: Darboux transformations
zbMATH Keywords: lattice Schrödinger equation
zbMATH Keywords: discrete spectrum
MaRDI profile type: Publication
full work available at URL: aphy.1995.1006
OpenAlex ID: W2093424215
DOI: 10.1006/APHY.1995.1006
description: scientific article; zbMATH DE number 727781