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(α) depending on a complex parameter α. If |α| ≠ 1 the spectrum of J(α) is discrete and formulas for eigenvalues and eigenvectors are established in terms of elliptic integrals and Jacobian elliptic functions. If |α| = 1, α ≠ ±1, the essential spectrum of J(α) 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(α) are obtained. Finally, the completeness of eigenvectors and Rodriguez-like formulas for orthogonal polynomials, studied previously by Carlitz, are proved.