LAUSR

We study semi-infinite Jacobi matrices H = H 0 + V corresponding to trace class perturbations V of the " free " discrete Schrodinger operator H 0. Our goal is to construct various spectral quantities of the operator H, such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair H 0 , H, the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials P n (z) associated to the Jacobi matrix H as n → ∞. In particular, we consider the case of z inside the spectrum [−1, 1] of H 0 when this asymptotics has an oscillating character of the Bernstein-Szego type and the case of z at the end points ±1.