Analytic scattering theory for Jacobi operators and Bernstein-Szeg\"o asymptotics of orthogonal polynomials

TL;DR

This paper develops an analytic scattering theory for Jacobi operators with trace class perturbations, deriving spectral quantities and asymptotics of associated orthogonal polynomials, including Bernstein-Szeg"o type behavior.

Contribution

It introduces a comprehensive spectral analysis framework for Jacobi matrices with trace class perturbations, linking spectral data to polynomial asymptotics.

Findings

Derived explicit asymptotics of orthogonal polynomials inside the spectrum.

Established spectral quantities such as wave operators and scattering matrix.

Identified Bernstein-Szeg"o type oscillatory behavior at spectrum endpoints.

Abstract

We study semi-infinite Jacobi matrices $H=H_{0}+V$ corresponding to trace class perturbations $V$ of the "free" discrete Schr\"odinger 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-Szeg\"o type and the case of $z$ at the end points $\pm 1$.