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arXiv · 1711.05029

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

D. R. Yafaev
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Abstract

We study semi-infinite Jacobi matrices $H=H_{0}+V$ corresponding to trace class perturbations $V$ of the "free" discrete Schrödinger 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 $n\to\infty$. In particular, we consider the case of $z$ inside the spectrum $[-1,1]$ of $H_{0}$ when this asymptotics has an oscillating character of the Bernstein-Szegö type and the case of $z$ at the end points $\pm 1$.

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BibTeXRIS

D. R. Yafaev. 2017-11-14. Analytic scattering theory for Jacobi operators and Bernstein-Szegö asymptotics of orthogonal polynomials. https://doi.org/10.1142/s0129055x18400196

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