Spectral Analysis Of Non-self-adjoint Jacobi Operator Associated With Jacobian Elliptic Functions

Petr Siegl, František Štampach

Research output: Contribution to journalArticle

2 Citations (Scopus)
10 Downloads (Pure)

Abstract

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.

Original languageEnglish
Pages (from-to)901-928
Number of pages28
JournalOperators and Matrices
Volume11
Issue number4
DOIs
Publication statusPublished - 01 Dec 2017
Externally publishedYes

Keywords

  • Jacobian elliptic functions
  • Non-self-adjoint Jacobi operator
  • Weyl m-function

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory

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