Analytic Scattering Theory for Jacobi Operators and Bernstein-Szego Asymptotics of Orthogonal Polynomials

被引:5
作者
Yafaev, D. R. [1 ,2 ]
机构
[1] Univ Rennes 1, IRMAR, Campus Beaulieu, F-35042 Rennes, France
[2] SPGU, Univ Nab 7-9, St Petersburg 199034, Russia
基金
俄罗斯科学基金会;
关键词
Jacobi matrices; discrete Schrodinger operator; orthogonal polynomials; asymptotics for large numbers; Szego function; MATRICES;
D O I
10.1142/S0129055X18400196
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
We study semi-infinite Jacobi matrices H = H-0 + V corresponding to trace class perturbations V of the "free" discrete Schrodinger operator H-0. 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 H-0, H, the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials P-n(z) associated to the Jacobi matrix H as n -> infinity. In particular, we consider the case of z inside the spectrum [-1, 1] of H-0 when this asymptotic has an oscillating character of the Bernstein-Szego type and the case of z at the end points +/- 1.
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页数:47
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