Spectral properties of operators using tridiagonalisation

TL;DR

This paper introduces a general method for tridiagonalising various operators using orthogonal polynomials, enabling spectral decomposition through their orthogonality measures, with applications to classical and q-analogues.

Contribution

It presents a unified scheme for tridiagonalising differential and q-difference operators via orthogonal polynomials, linking spectral theory with special functions.

Findings

Derived spectral decompositions for specific differential and q-difference operators.

Established connections between the tridiagonal form and classical orthogonal polynomials.

Extended the Jacobi function transform to a q-analogue for certain operators.

Abstract

A general scheme for tridiagonalising differential, difference or q-difference operators using orthogonal polynomials is described. From the tridiagonal form the spectral decomposition can be described in terms of the orthogonality measure of generally different orthogonal polynomials. Three examples are worked out: (1) related to Jacobi and Wilson polynomials for a second order differential operator, (2) related to little q-Jacobi polynomials and Askey-Wilson polynomials for a bounded second order q-difference operator, (3) related to little q-Jacobi polynomials for an unbounded second order q-difference operator. In case (1) a link with the Jacobi function transform is established, for which we give a q-analogue using example (2).