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

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.