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 $|\alpha|\neq1$ 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 $|\alpha|=1$, $\alpha \neq \pm 1$, 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. 
Petr Siegl and Franti\v{s}ek \v{S}tampach
math.SP math-ph math.FA math.MP