Periodic perturbations of unbounded Jacobi matrices III: The soft edge regime

TL;DR

This paper provides a detailed spectral analysis of Jacobi matrices with periodic modulations at the soft edge, revealing conditions for self-adjointness, discreteness of spectrum, and absolute continuity, along with a formula for spectral density.

Contribution

It offers new insights into the spectral properties of periodically modulated Jacobi matrices at the soft edge, including self-adjointness and spectral type classifications.

Findings

Operators are always self-adjoint regardless of the modulated sequence.

Superlinear growth of the sequence leads to a discrete spectrum.

Conditions are identified for absolute continuity of the spectrum.

Abstract

We present pretty detailed spectral analysis of Jacobi matrices with periodically modulated entries in the case when $0$ lies on the soft edge of the spectrum of the corresponding periodic Jacobi matrix. In particular, we show that the studied operators are always self-adjoint irrespective of the modulated sequence. Moreover, if the growth of the modulated sequence is superlinear, then the spectrum of the considered operators is always discrete. Finally, we study regular perturbations of this class in the linear and the sublinear cases. We impose conditions assuring that the spectrum is absolute continuous on some regions of the real line. A constructive formula for the density in terms of Tur\'an determinants is also provided.