Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials

Abstract

We find and discuss asymptotic formulas for orthonormal polynomials Pn(z) with recurrence coefficients an, bn. Our main goal is to consider the case where off-diagonal elements an → ∞ as n → ∞. Formulas obtained are essentially different for relatively small and large diagonal elements bn. Our analysis is intimately linked with spectral theory of Jacobi operators J with coefficients an, bn and a study of the corresponding second order difference equations. We introduce the Jost solutions fn(z), n ≥ −1, of such equations by a condition for n → ∞ 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 Pn(z) by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for Pn(z) as n → ∞ in terms of the Wronskian of the solutions Pn(z) and fn(z). The formulas obtained for Pn(z) generalize the asymptotic formulas for the classical Hermite polynomials where an = √ (n+ 1)/2 and bn = 0. The spectral structure of Jacobi operators J depends crucially on a rate of growth of the off-diagonal elements an as n → ∞. If the Carleman condition is satisfied, which, roughly speaking, means that an = O(n), and the diagonal elements bn are small compared to an, then J 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 |f −1(λ ± i0)| of the Jost solutions. On the contrary, if the Carleman condition is violated, then the spectrum of J is discrete. We also review the case of stabilizing recurrence coefficients when an tend to a positive constant and bn → 0 as n → ∞. It turns out that the cases of stabilizing and increasing recurrence coefficients can be treated in an essentially same way.