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

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.