Spectral analysis of two doubly infinite Jacobi matrices with exponential entries

Abstract

We provide a complete spectral analysis of all self-adjoint operators acting on ${\ell }^2(\mathbb{Z})$ which are associated with two doubly infinite Jacobi matrices with entries given by $$q^{-n+1}{\delta }_{m,n-1}+q^{-n}{\delta }_{m,n+1}$$ and $${\delta }_{m,n-1}+\alpha q^{-n}{\delta }_{m,n}+{\delta }_{m,n+1},$$ respectively, where $q\in (0,1)$ and $\alpha \in \mathbb{R}$. As an application, we derive orthogonality relations for the Ramanujan entire function and the third Jackson $q$-Bessel function.

22 pages, preprint