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

TL;DR

This paper analyzes the spectral properties of a family of Jacobi operators depending on a complex parameter, revealing discrete spectra, essential spectra, and orthogonal polynomial relations using elliptic functions.

Contribution

It provides explicit formulas for eigenvalues, eigenvectors, and the Weyl m-function of Jacobi operators linked to Jacobian elliptic functions, including spectral characterizations and orthogonal polynomial formulas.

Findings

Discrete spectrum for || eq 1 with explicit formulas

Essential spectrum covers entire complex plane when ||=1,  eq \u00b1 1

Established completeness of eigenvectors and Rodriguez-like formulas

Abstract

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.