Semiclassical asymptotic behavior of orthogonal polynomials

TL;DR

This paper develops spectral theory for Jacobi operators with slowly stabilizing coefficients to derive asymptotic formulas for orthonormal polynomials, generalizing classical results in the semiclassical regime.

Contribution

It introduces a spectral approach and an Ansatz for solutions of the difference equation, extending classical asymptotics to long-range coefficient cases.

Findings

Derived asymptotic formulas for orthogonal polynomials with slowly stabilizing recurrence coefficients.

Developed spectral theory for Jacobi operators with long-range coefficients.

Generalized Bernstein-Szeg"o asymptotic formulas in the semiclassical setting.

Abstract

Our goal is to find asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with the recurrence coefficients slowly stabilizing as $n\to\infty$. To that end, we develop spectral theory of Jacobi operators with long-range coefficients and study the corresponding second order difference equation. We suggest an Ansatz for its solutions playing the role of the semiclassical Green-Liouville Ansatz for solutions of the Schr\"odinger equation. The formulas obtained for $P_{n}(z)$ as $n\to\infty$ generalize the classical Bernstein-Szeg\"o asymptotic formulas.