Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials

Abstract

We find and discuss asymptotic formulas for orthonormal polynomials [Formula: see text] with recurrence coefficients [Formula: see text]. Our main goal is to consider the case where off-diagonal elements [Formula: see text] as [Formula: see text]. Formulas obtained are essentially different for relatively small and large diagonal elements [Formula: see text]. Our analysis is intimately linked with spectral theory of Jacobi operators [Formula: see text] with coefficients [Formula: see text] and a study of the corresponding second order difference equations. We introduce the Jost solutions [Formula: see text], [Formula: see text], of such equations by a condition for [Formula: see text] and suggest an Ansatz for them playing the role of the semiclassical Liouville–Green Ansatz for solutions of the Schrödinger equation. This allows us to study the spectral structure of Jacobi operators and their eigenfunctions [Formula: see text] by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for [Formula: see text] as [Formula: see text] in terms of the Wronskian of the solutions [Formula: see text] and [Formula: see text]. The formulas obtained for [Formula: see text] generalize the asymptotic formulas for the classical Hermite polynomials where [Formula: see text] and [Formula: see text]. The spectral structure of Jacobi operators [Formula: see text] depends crucially on a rate of growth of the off-diagonal elements [Formula: see text] as [Formula: see text]. If the Carleman condition is satisfied, which, roughly speaking, means that [Formula: see text], and the diagonal elements [Formula: see text] are small compared to [Formula: see text], then [Formula: see text] has the absolutely continuous spectrum covering the whole real axis. We obtain an expression for the corresponding spectral measure in terms of the boundary values [Formula: see text] of the Jost solutions. On the contrary, if the Carleman condition is violated, then the spectrum of [Formula: see text] is discrete. We also review the case of stabilizing recurrence coefficients when [Formula: see text] tend to a positive constant and [Formula: see text] as [Formula: see text]. It turns out that the cases of stabilizing and increasing recurrence coefficients can be treated in an essentially same way.