The spectral measure of a Jacobi matrix in terms of the Fourier transform of the perturbation
type: Item
instance of: scholarly article
title: The spectral measure of a Jacobi matrix in terms of the Fourier transform of the perturbation
zbMATH Open document ID: 1066.47030
DOI: 10.1007/BF02385486
author: Oleg Safronov
published in: Arkiv för Matematik
publication date: 2004-09-28T00:00:00Z
review text: For the last few years, there has been a splash of activity around the study of spectral properties of Schrödinger operators and Jacobi matrices initiated by the seminal paers of Deift-Killip, Killip-Simon and Denisov. The present paper is another one in a row. The author relates the properties of matrix entries and the correspondind spectral measure \(\mu\), more precisely, the convergence of the integral of \(\log(1/\mu'(2\cos t))\) with respect to the weight \(\rho\) being an arbitrary (in a sense) positive trigonometric polynomial. The main result claims that whenever \(J-J_0\) belongs to the Schatten class \(\mathcal{S}_3\), such integrability is equivalent to the following three conditions: the standard structure of supp \(\mu\), the quasi-Szegő condition with respect to \(\rho\) and the Lieb-Thirring bound.
Mathematics Subject Classification ID: 47B36
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 34C40
Mathematics Subject Classification ID: 81U10
zbMATH DE Number: 2103855
zbMATH Keywords: Jacobi matrices
zbMATH Keywords: spectral measures
zbMATH Keywords: Schatten classes
zbMATH Keywords: quasi-Szegő condition
zbMATH Keywords: Lieb-Thirring bound
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
cites work: Q5508534
cites work: Orthogonal polynomials from the viewpoint of scattering theory
cites work: Orthogonal polynomials. II
cites work: Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results
cites work: The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials
cites work: On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials
cites work: On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some Sturm--Liouville operators with square summable potential
cites work: Solitons. Mathematical methods for physicists
cites work: On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators
cites work: Jacobi matrices with absolutely continuous spectrum
cites work: Q4781494
cites work: Sum rules for Jacobi matrices and their applications to spectral theory
cites work: Absolutely continuous spectra of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials
cites work: Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators
cites work: Asymptotics of the Fourier transform of the spectral measure for Schr$ouml$dinger operators with bounded and unbounded sparse potentials
cites work: On new relations between spectral properties of Jacobi matrices and their coefficients
cites work: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators
cites work: First KdV integrals and absolutely continuous spectrum for 1-D Schrödinger operator
cites work: Simultaneous interpolation and approximation of continuous functions
cites work: The absolute continuous spectrum of one-dimensional Schrödinger operators with decaying potentials
cites work: Some Jacobi matrices with decaying potential and dense point spectrum
cites work: Some Schrödinger operators with dense point spectrum
cites work: Sum rules and the Szegő condition for orthogonal polynomials on the real line
description: scientific article; zbMATH DE number 2103855