Resonances for vector‐valued Jacobi operators on half‐lattice

Evgeny Korotyaev

Mathematische Nachrichten, 2025, vol. 298, issue 9, 3075-3113

Abstract: We study resonances for Jacobi operators on the half lattice with matrix‐valued coefficients and finitely supported perturbations. We describe a forbidden domain, the geometry of resonances and their asymptotics when the main coefficient of the perturbation (determining its length of the support) goes to zero. Moreover, we show that Any sequence of points on the complex plane can be resonances for some Jacobi operator. In particular, the multiplicity of a resonance can be any number. The Jost determinant coincides with the Fredholm determinant up to the constant. The S‐matrix on the a.c spectrum determines the perturbation uniquely. The value of the Jost matrix at any finite sequence of points on the a.c spectrum determine the Jacobi matrix uniquely. The length of this sequence is equal to the upper point of the support perturbation.

Date: 2025
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