We study semi-infinite Jacobi matrices H=H0+V corresponding to trace class perturbations V of the “free” discrete Schrödinger operator H0. 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 H0, H, the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials Pn(z) associated to the Jacobi matrix H as n→∞. In particular, we consider the case of z inside the spectrum [−1,1] of H0 when this asymptotic has an oscillating character of the Bernstein–Szegö type and the case of z at the end points ±1.