2022/02/04 by D. R. Yafaev, Yafaev, D. R.
Mathematics · Physics and Astronomy · #33C45 #39A70 #47A40 #47B39 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · doi:10.48550/arxiv.2202.02087
openalex publication_date 2022/02/04 · openalex created_date 2022/07/31 · openalex updated_date 2026/07/28
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.