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

TL;DR

This paper establishes exponential decay bounds for eigenfunctions of unbounded Jacobi matrices, providing sharp results and criteria for spectral properties, including the existence of spectral mobility edges.

Contribution

It introduces new decay bounds for eigenfunctions of unbounded Jacobi matrices and applies these to criteria for spectral types and mobility edges.

Findings

Bounds on eigenfunction decay are sharp.

Criteria for eigenfunction existence in spectral gaps.

Examples of matrices with spectral mobility edges.

Abstract

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.