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

TL;DR

This paper investigates a class of unbounded Jacobi matrices that exhibit a first-order phase transition, analyzing spectral types and asymptotics at the critical transition point.

Contribution

It determines the spectral type and asymptotic behavior of solutions at the phase transition point for a specific class of unbounded Jacobi matrices.

Findings

Spectral type changes from discrete to absolutely continuous at the transition.

Asymptotic solutions are characterized at the critical point.

The phase transition is confirmed to be first-order.

Abstract

We consider a class of Jacobi matrices with unbounded coefficients. This class is known to exhibit a first-order phase transition in the sense that, as a parameter is varied, one has purely discrete spectrum below the transition point and purely absolutely continuous spectrum above the transition point. We determine the spectral type and solution asymptotics at the transition point.