The spectral measure of a Jacobi matrix in terms of the Fourier transform of the perturbation
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.
scientific article; zbMATH DE number 2103855