Spectral analysis of two doubly infinite Jacobi matrices with exponential entries

TL;DR

This paper conducts a comprehensive spectral analysis of two specific doubly infinite Jacobi matrices with exponential entries, revealing their spectral properties and deriving orthogonality relations for special functions.

Contribution

It provides a complete spectral characterization of two classes of Jacobi matrices with exponential entries and connects these results to orthogonality relations of special functions.

Findings

Spectral properties of the two Jacobi matrices are fully characterized.

Orthogonality relations for Ramanujan entire function are derived.

Orthogonality relations for the third Jackson q-Bessel function are established.

Abstract

We provide a complete spectral analysis of all self-adjoint operators acting on $\ell^{2}(\mathbb{Z})$ which are associated with two doubly infinite Jacobi matrices with entries given by $$ q^{-n+1}\delta_{m,n-1}+q^{-n}\delta_{m,n+1} $$ and $$ \delta_{m,n-1}+\alpha q^{-n}\delta_{m,n}+\delta_{m,n+1}, $$ respectively, where $q\in(0,1)$ and $\alpha\in\mathbb{R}$. As an application, we derive orthogonality relations for the Ramanujan entire function and the third Jackson $q$-Bessel function.