Spectral Analysis of Certain Schr\"odinger Operators

TL;DR

This paper extends the $J$-matrix method to various operators, analyzing their spectra and eigenfunctions, and introduces new orthogonal polynomials with explicit recurrence relations in discrete quantum models.

Contribution

It generalizes the $J$-matrix method to difference and $q$-difference operators and discovers new orthogonal polynomials with explicit properties.

Findings

Explicit spectral measures for several operators

New orthogonal polynomials with known recurrence relations

Eigenfunction expansions for discrete quantum models

Abstract

The $J$-matrix method is extended to difference and $q$-difference operators and is applied to several explicit differential, difference, $q$-difference and second order Askey-Wilson type operators. The spectrum and the spectral measures are discussed in each case and the corresponding eigenfunction expansion is written down explicitly in most cases. In some cases we encounter new orthogonal polynomials with explicit three term recurrence relations where nothing is known about their explicit representations or orthogonality measures. Each model we analyze is a discrete quantum mechanical model in the sense of Odake and Sasaki [J. Phys. A: Math. Theor. 44 (2011), 353001, 47 pages].