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

Abstract

We consider self-adjoint unbounded Jacobi matrices with diagonal q_n=n and weights λ_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.

References in corpus (1)

• The Case of Critical Coupling in a Class of Unbounded Jacobi Matrices Exhibiting a First-Order Phase Transition

Cited by in corpus (6)

• Periodic perturbations of unbounded Jacobi matrices III: The soft edge regime
• About essential spectra of unbounded Jacobi matrices
• Asymptotic behaviour of Christoffel-Darboux kernel via three-term recurrence relation II
• Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case II
• Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case
• Asymptotic zeros' distribution of orthogonal polynomials with unbounded recurrence coefficients