Analytic scattering theory for Jacobi operators and Bernstein-Szegö asymptotics of orthogonal polynomials

Abstract

We study semi-infinite Jacobi matrices $H=H_0+V$ corresponding to trace class perturbations $V$ of the "free" discrete Schrödinger 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 $n\to \infty$. 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-Szegö type and the case of $z$ at the end points $\pm 1$.

Dedicated to the memory of Lyudvig Dmitrievich Faddeev

References in corpus (1)

• A point interaction for the discrete Schrödinger operator and generalized Chebyshev polynomials