Spectral analysis of two doubly infinite Jacobi matrices with\n exponential entries

2016/05/02 by Mourad E. H. Ismail, Ismail, Mourad E. H., František Štampach +1
Mathematics · Physics and Astronomy · #33D90 #47B36 #FOS: Mathematics #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1605.00393

openalex publication_date 2016/05/02 · openalex created_date 2022/08/22 · openalex updated_date 2026/07/28

Abstract

We provide a complete spectral analysis of all self-adjoint operators acting\non \ℓ²(\ℤ) which are associated with two doubly infinite Jacobi\nmatrices with entries given by q^-n+1
delta_m,n-1+q^-n
delta_m,n+1\n and
delta_m,n-1+
alpha q^-n
delta_m,n+
delta_m,n+1,\nrespectively, where q\∈(0,1) and \α\∈\ℝ. As an application,\nwe derive orthogonality relations for the Ramanujan entire function and the\nthird Jackson q-Bessel function.\n

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