The Szego Condition for Coulomb Jacobi Matrices

Abstract

A Jacobi matrix with $a_n\to 1$, $b_n\to 0$ and spectral measure ${\nu }^{\prime }(x)dx+d{\nu }_{sing}(x)$ satisfies the Szeg\H o condition if ${\int }_0^{\pi }\ln [{\nu }^{\prime }(2\cos \theta )]d\theta$ is finite. We prove that if $a_n = 1 + \frac α{n} + O(n^{-1-\eps})$ and $b_n = \frac β{n} + O(n^{-1-\eps})$ with $2\alpha \ge |\beta |$ and $\eps>0$, then the corresponding matrix is Szeg\H o.