Semiclassical asymptotic behavior of orthogonal polynomials

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 Schrödinger equation. The formulas obtained for $P_n(z)$ as $n\to \infty$ generalize the classical Bernstein-Szegö asymptotic formulas.

Cited by in corpus (1)

• Asymptotic behavior of orthogonal polynomials without the Carleman condition