Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices

Details Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices

2007-11-26

arxiv.org/abs/0711.4035v2

Mathematics

Spectral Theory

This is a substantially revised and expanded version of 0711.4035v1

Scientific paper

Bounds on the exponential decay of generalized eigenfunctions of bounded and unbounded selfadjoint Jacobi matrices are established. Two cases are considered separately: (i) the case in which the spectral parameter lies in a general gap of the spectrum of the Jacobi matrix and (ii) the case of a lower semi-bounded Jacobi matrix with values of the spectral parameter below the spectrum. It is demonstrated by examples that both results are sharp. We apply these results to obtain a "many barriers-type" criterion for the existence of square-summable generalized eigenfunctions of an unbounded Jacobi matrix at almost every value of the spectral parameter in suitable open sets. As an application, we provide examples of unbounded Jacobi matrices with a spectral mobility edge.

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