Spectral density in a Moszynski's class of Jacobi matrices. Spectral phase transition of 2nd type

Abstract

In this paper it is considered a spectral density for a class of Jacobi matrices with absolutely continuous spectrum that was examined first by Moszynski. It is shown that the corresponding spectral density is equivalent to the positive continuous function everywhere except the point $x=0$. In the point $x=0$ the spectral density may be finite as well as essentially infinite depending on the parameter. So in this class we discover an example of 2nd type spectral phase transition.