TY - JOUR

T1 - Analytic scattering theory for Jacobi operators and Bernstein-Szego asymptotics of orthogonal polynomials

AU - Yafaev, D. R.

PY - 2018

Y1 - 2018

N2 - We study semi-infinite Jacobi matrices H=H0+V corresponding to trace class perturbations V of the “free” discrete Schrödinger operator H0. Our goal is to construct various spectral quantities of the operator H, such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair H0, H, the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials Pn(z) associated to the Jacobi matrix H as n→∞. In particular, we consider the case of z inside the spectrum [−1,1] of H0 when this asymptotic has an oscillating character of the Bernstein–Szegö type and the case of z at the end points ±1.

AB - We study semi-infinite Jacobi matrices H=H0+V corresponding to trace class perturbations V of the “free” discrete Schrödinger operator H0. Our goal is to construct various spectral quantities of the operator H, such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair H0, H, the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials Pn(z) associated to the Jacobi matrix H as n→∞. In particular, we consider the case of z inside the spectrum [−1,1] of H0 when this asymptotic has an oscillating character of the Bernstein–Szegö type and the case of z at the end points ±1.

KW - Jacobi matrices

KW - discrete Schrödinger operator

KW - orthogonal polynomials

KW - Asymptotics for large numbers

KW - Szegö function

UR - https://proxy.library.spbu.ru:2841/doi/10.1142/S0129055X18400196

M3 - Article

VL - 30

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

SN - 0129-055X

IS - 8

M1 - 1840019

ER -