Discrete models of the harmonic oscillator and a discrete analogue of Gauss' hypergeometric equation
type: Item
instance of: scholarly article
title: Discrete models of the harmonic oscillator and a discrete analogue of Gauss' hypergeometric equation
zbMATH Open document ID: 1003.33003
DOI: 10.1023/A:1012922627100
published in: The Ramanujan Journal
publication date: 2002-06-19T00:00:00Z
review text: A bispectral problem is an eigenvalue problem of the form \[ (Af)(x,y)=\lambda(y) f(x,y),\qquad (Bf)(x,y)=\mu(x) f(x,y), \] where \(A\) is a linear operator acting on the \(x\)-variable and \(B\) acts on the \(y\)-variable. In the original formulation of the problem [\textit{H. Duistermaat} and \textit{F. A. Grünbaum}, Commun. Math. Phys. 103, 177-240 (1986; Zbl 0625.34007)] \(A\) is a Schrödinger operator and \(B\) an arbitrary linear ordinary differential operator. Recently, cases when one or both operators are difference operators have received much attention. In [\textit{F. A. Grünbaum} and \textit{L. Haine}, Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 26, 143-172 (1997; Zbl 0868.35116)] the case when \(A\) is a tridiagonal difference operator and \(B\) a second order differential operator was considered. It was shown that the solutions may be described in terms of solutions to Gauss's hypergeometric differential equation. In the present paper, both \(A\) and \(B\) are tridiagonal matrices and the spectrum of \(A\) is assumed to be linear. That is, the problem \[ a_{i+1} f_{i+1,j}+a_i f_{ij}+a_{i-1} f_{i-1,j}=j f_{ij}, \] \[ b_{j+1} f_{i,j+1}+b_j f_{ij}+b_{j-1}f _{i,j-1} =\mu_i f_{ij} \] with \(f\) a doubly infinite matrix is considered. It is shown that \(\mu_i\) must be a second order polynomial in \(i\), and that the remaining data may be described in terms of the solutions to a difference equation of the form \[ w_{j+1}+p_2(j)w_j+p_4(j)w_{j-1}=0, \] with \(p_k\) polynomials of degree \(k\) of a certain, rather complicated, form. The latter equation is viewed as a discrete analogue of Gauss's hypergeometric equation.
reviewed by: Hjalmar Rosengren
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 1757281
zbMATH Keywords: bispectral problem
zbMATH Keywords: discrete hypergeometric equation
author: F. Alberto Grünbaum
MaRDI profile type: Publication
full work available at URL: a:1012922627100
OpenAlex ID: W57383650
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description: scientific article; zbMATH DE number 1757281