2014/04/30 by František Štampach, Stampach, Frantisek, P. Šťovı́ček +1
Computer Science · Mathematics · Physics and Astronomy · #33D45 #39A70 #47A10 #47B36 #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1404.7647
openalex publication_date 2014/04/30 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
A family \T(\ν), \ν\∈\ℝ, of semiinfinite positive\nJacobi matrices is introduced with matrix entries taken from the Hahn-Exton\nq-difference equation. The corresponding matrix operators defined on the\nlinear hull of the canonical basis in \ℓ2(\ℤ+) are\nessentially self-adjoint for |\ν|\≥1 and have deficiency indices (1,1)\nfor |\ν|<1. A convenient description of all self-adjoint extensions is\nobtained and the spectral problem is analyzed in detail. The spectrum is\ndiscrete and the characteristic equation on eigenvalues is derived explicitly\nin all cases. Particularly, the Hahn-Exton q-Bessel function J\ν(z;q)\nserves as the characteristic function of the Friedrichs extension. As a direct\napplication one can reproduce, in an alternative way, some basic results about\nthe q-Bessel function due to Koelink and Swarttouw.\n