Analytic Scattering Theory for Jacobi Operators and Bernstein–Szegö Asymptotics of Orthogonal Polynomials

We study semi-infinite Jacobi matrices [Formula: see text] corresponding to trace class perturbations [Formula: see text] of the “free” discrete Schrödinger operator [Formula: see text]. Our goal is to construct various spectral quantities of the operator [Formula: see text], such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair [Formula: see text], [Formula: see text], the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials [Formula: see text] associated to the Jacobi matrix [Formula: see text] as [Formula: see text]. In particular, we consider the case of [Formula: see text] inside the spectrum [Formula: see text] of [Formula: see text] when this asymptotic has an oscillating character of the Bernstein–Szegö type and the case of [Formula: see text] at the end points [Formula: see text].