A first-order spectral phase transition in a class of periodically modulated Hermitian Jacobi matrices

Abstract

We consider self-adjoint unbounded Jacobi matrices with diagonal $q_n = b_{n}n$ and off-diagonal entries $\lambda_n = n$, where bn is a $2$-periodical sequence of real numbers. The parameter space is decomposed into several separate regions, where the spectrum of the operator is either purely absolutely continuous or discrete. We study the situation where the spectral phase transition occurs, namely the case of $b_{1}b_{2} = 4$. The main motive of the paper is the investigation of asymptotics of generalized eigenvectors of the Jacobi matrix. The pure point part of the spectrum is analyzed in detail.