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

Petr Siegl; František Štampach

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arXiv · 1603.01052

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

Petr Siegl·František Štampach

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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.

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BibTeX RIS

Petr Siegl, František Štampach. 2016-03-03. Spectral analysis of non-self-adjoint Jacobi operator associated with Jacobian elliptic functions. https://arxiv.org/abs/1603.01052

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