TY - JOUR

T1 - Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials

AU - Yafaev, D. R.

N1 - Publisher Copyright: © 2022 The Author(s).

PY - 2022/12/1

Y1 - 2022/12/1

N2 - 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.

AB - 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.

KW - Jacobi matrices

KW - increasing recurrence coefficients

KW - difference equations

KW - orthogonal polynomials

KW - Asymptotics for large numbers

KW - asymptotics for large numbers

UR - https://www.worldscientific.com/doi/abs/10.1142/S1664360722500023

UR - http://www.scopus.com/inward/record.url?scp=85135404382&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/4d3e3526-992b-362f-891a-0d9b56ff8501/

U2 - 10.1142/S1664360722500023

DO - 10.1142/S1664360722500023

M3 - Article

VL - 12

JO - Bulletin of Mathematical Sciences

JF - Bulletin of Mathematical Sciences

SN - 1664-3607

IS - 3

M1 - 2250002

ER -