Spectral phase transitions for Jacobi matrices and asymptotic properties of orthogonal polynomials. Some examples

The talk is devoted to phenomena of spectral phase transitions for families of Jacobi matrices. Relations between spectral properties of Jacobi matrices and asymptotic properties of corresponding orthonormal polynomials will be discussed, and several examples will be given. In particular, we will consider the discrete Schroedinger operator with Wigner-von Neumann potential and formulate a recent result about the asymptotics of its spectral density which was obtained by analyzing the behavior of orthonormal polynomials.