Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials

TL;DR

This paper develops asymptotic formulas for orthogonal polynomials associated with Jacobi operators, especially when off-diagonal elements grow unbounded, linking spectral theory and difference equations to generalize classical polynomial asymptotics.

Contribution

It introduces new asymptotic formulas for orthogonal polynomials with unbounded recurrence coefficients, using spectral theory and Jost solutions, extending classical Hermite polynomial results.

Findings

Asymptotic formulas depend on the size of diagonal elements $b_{n}$.

Introduction of Jost solutions for second order difference equations.

Generalization of Hermite polynomial asymptotics to unbounded coefficient cases.

Abstract

We find and discuss asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with recurrence coefficients $a_{n}, b_{n}$. Our main goal is to consider the case where off-diagonal elements $a_{n}\to\infty$ as $n\to\infty$. Formulas obtained are essentially different for relatively small and large diagonal elements $b_{n}$. Our analysis is intimately linked with spectral theory of Jacobi operators $J$ with coefficients $a_{n}, b_{n}$ and a study of the corresponding second order difference equations. We introduce the Jost solutions $f_{n}(z)$, $n\geq -1$, of such equations by a condition for $n\to\infty$ and suggest an Ansatz for them playing the role of the semiclassical Liouville-Green Ansatz for solutions of the Schr\"odinger equation. This allows us to study the spectral structure of Jacobi operators and their eigenfunctions $P_{n}(z)$ by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for $P_{n}(z)$ as $n \to\infty$ in terms of the Wronskian of the solutions $ P_{n} (z) $ and $ f_{n} (z)$. The formulas obtained for $P_{n}(z)$ generalize the asymptotic formulas for the classical Hermite polynomials where $a_{n}=\sqrt{(n+1)/2}$ and $b_{n}=0$.