An example of spectral phase transition phenomenon in a class of Jacobi matrices with periodically modulated weights

TL;DR

This paper investigates spectral phase transitions in a class of Jacobi matrices with periodically modulated weights, revealing regions of purely absolutely continuous or discrete spectra and analyzing transition lines.

Contribution

It provides a detailed analysis of spectral phase transitions in Jacobi matrices with periodic weights, including asymptotic eigenvector behavior and degenerate cases.

Findings

Spectral phase transition occurs at specific parameter values.

Absolutely continuous spectrum covers either (-∞, 1/2) or (1/2, ∞).

Degenerate cases also exhibit phase transition phenomena.

Abstract

We consider self-adjoint unbounded Jacobi matrices with diagonal q_n=n and weights \lambda_n=c_n n, where c_n is a 2-periodical sequence of real numbers. The parameter space is decomposed into several separate regions, where the spectrum is either purely absolutely continuous or discrete. This constitutes an example of the spectral phase transition of the first order. We study the lines where the spectral phase transition occurs, obtaining the following main result: either the interval (-\infty;1/2) or the interval (1/2;+\infty) is covered by the absolutely continuous spectrum, the remainder of the spectrum being pure point. The proof is based on finding asymptotics of generalized eigenvectors via the Birkhoff-Adams Theorem. We also consider the degenerate case, which constitutes yet another example of the spectral phase transition.