Semiclassical asymptotic behavior of orthogonal polynomials

Our goal is to find asymptotic formulas for orthonormal polynomials P n ( z ) with the recurrence coefficients slowly stabilizing as n → ∞ . To that end, we develop scattering theory of Jacobi operators with long-range coefficients and study the corresponding second-order difference equation. We introduce the Jost solutions f n ( z ) of this equation by a condition for n → ∞ and suggest an Ansatz for them playing the role of the semiclassical Liouville–Green Ansatz for the corresponding solutions of the Schrödinger equation. This allows us to study 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 → ∞ in terms of the Wronskian of the solutions { P n ( z ) } and { f n ( z ) } .