Discrete spectrum in a critical coupling case of Jacobi matrices with spectral phase transitions by uniform asymptotic analysis

TL;DR

This paper investigates the spectral properties of a family of Jacobi matrices at phase transition boundaries, establishing spectrum discreteness through a novel uniform asymptotic analysis of eigenvectors.

Contribution

It introduces a new method for uniform asymptotic analysis of eigenvectors applicable to Jacobi matrices at spectral phase transitions.

Findings

Spectrum is discrete on the positive real axis at transition boundaries.

Developed a versatile technique for asymptotic analysis of eigenvectors.

Applicable to a broad class of Jacobi matrices.

Abstract

For a two-parameter family of Jacobi matrices exhibiting first-order spectral phase transitions, we prove discreteness of the spectrum in the positive real axis when the parameters are in one of the transition boundaries. To this end we develop a method for obtaining uniform asymptotics, with respect to the spectral parameter, of the generalized eigenvectors. Our technique can be applied to a wide range of Jacobi matrices.