Analytic Scattering Theory for Jacobi Operators and Bernstein-Szego Asymptotics of Orthogonal Polynomials

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 -> infinity. In particular, we consider the case of z inside the spectrum [-1, 1] of H-0 when this asymptotic has an oscillating character of the Bernstein-Szego type and the case of z at the end points +/- 1.