Spectral analysis of non-self-adjoint Jacobi operator associated with Jacobian elliptic functions

We perform the spectral analysis of a family of Jacobi operators $J(\alpha )$ depending on a complex parameter $\alpha$. If $\mathrm{\mid }\alpha \mathrm{\mid }\mathrm{\ne }1$ the spectrum of $J(\alpha )$ is discrete and formulas for eigenvalues and eigenvectors are established in terms of elliptic integrals and Jacobian elliptic functions. If $\mathrm{\mid }\alpha \mathrm{\mid }=1$, $\alpha \mathrm{\ne }\pm 1$, the essential spectrum of $J(\alpha )$ covers the entire complex plane. In addition, a formula for the Weyl $m$-function as well as the asymptotic expansions of solutions of the difference equation corresponding to $J(\alpha )$ are obtained. Finally, the completeness of eigenvectors and Rodriguez-like formulas for orthogonal polynomials, studied previously by Carlitz, are proved.