Bernstein-Szegö asymptotique des polynômes orthogonaux

We study semi-infinite Jacobi matrices \(H=H_{0}+V\) corresponding to trace class perturbations \(V\) of the "free" discrete Schrödinger operator \(H_{0}\) and properties of the associated orthonormal polynomials \(P_{n}(z)\). 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   polynomials \(P_{n}(z)\) as \(n\to\infty\) and gives a new look on the Bernstein-Szegö formulas. We give a proof of these formulas under essentially more general circumstances than in the original papers.