Semiclassical Asymptotic Behavior of Orthogonal Polynomials

Дмитрий Яфаев (Universite de Rennes, France & Saint Petersburg State University, Russia)
Thursday, 07 November 2019
16:00
конференц-зал, ауд.401 Moscow center for continuous mathematical education

Our goal is to find asymptotic formulas for orthonormal polynomials Pn(z) with the recurrence coefficients slowly stabilizing as n → ∞. 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 Pn(z) as n → ∞ generalize the classical Bernstein-Szeg ̈o asymptotic formulas.