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

Abstract

A family T(ν), ν ∈ ℝ, of semiinfinite positive Jacobi matrices is introduced with matrix entries takenfrom the Hahn-Exton q-difference equation. The corresponding matrix operators defined on the linear hullof the canonical basis in ℓ2(ℤ+) are essentially self-adjoint for |ν| ≥ 1 and have deficiency indices (1, 1) for|ν| < 1. A convenient description of all self-adjoint extensions is obtained and the spectral problem is analyzedin detail. The spectrum is discrete and the characteristic equation on eigenvalues is derived explicitlyin all cases. Particularly, the Hahn-Exton q-Bessel function Jν(z; q) serves as the characteristic function ofthe Friedrichs extension. As a direct application one can reproduce, in an alternative way, some basic resultsabout the q-Bessel function due to Koelink and Swarttouw.

Keywords

• jacobi matrix
• hahn-exton q-bessel function
• self-adjoint extension
• spectral problem