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

Siegl, Petr; Stampach, Frantisek (2017). Spectral analysis of non-self-adjoint Jacobi operator associated with Jacobian elliptic functions. Operators and Matrices, 11(4), pp. 901-928. Element 10.7153/oam-2017-11-64

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We perform the spectral analysis of a family of Jacobi operators J(α) depending on a complex parameter α. If ӀαӀ ≠ 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 ӀαӀ = 1, α ≠ ±1 the essential spectrum of J(α) covers the entire complex plane. In addition, a formula for the Weyl -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.

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