The Hahn-Exton $q$-Bessel function as the characteristic function of a Jacobi matrix

TL;DR

This paper introduces a family of Jacobi matrices linked to the Hahn-Exton q-Bessel function, analyzing their spectral properties and showing the q-Bessel function as the characteristic function of a specific matrix extension.

Contribution

It provides a new operator-theoretic framework connecting the Hahn-Exton q-Bessel function with Jacobi matrices and explicitly characterizes their spectral properties.

Findings

Spectrum is discrete for these matrices.

The Hahn-Exton q-Bessel function acts as the characteristic function of the Friedrichs extension.

All self-adjoint extensions are explicitly described.

Abstract

A family $\mathcal{T}^{(\nu)}$, $\nu\in\mathbb{R}$, of semiinfinite positive Jacobi matrices is introduced with matrix entries taken from the Hahn-Exton $q$-difference equation. The corresponding matrix operators defined on the linear hull of the canonical basis in $\ell^{2}(\mathbb{Z}_{+})$ are essentially self-adjoint for $|\nu|\geq1$ and have deficiency indices $(1,1)$ for $|\nu|<1$. A convenient description of all self-adjoint extensions is obtained and the spectral problem is analyzed in detail. The spectrum is discrete and the characteristic equation on eigenvalues is derived explicitly in all cases. Particularly, the Hahn-Exton $q$-Bessel function $J_{\nu}(z;q)$ serves as the characteristic function of the Friedrichs extension. As a direct application one can reproduce, in an alternative way, some basic results about the $q$-Bessel function due to Koelink and Swarttouw.