The Szego Condition for Coulomb Jacobi Matrices

Abstract

A Jacobi matrix with $a_n\to 1$, $b_n\to 0$ and spectral measure $ν'(x)dx + dν_{sing}(x)$ satisfies the Szeg\H o condition if $\int_{0}^π\ln \bigl[ ν'(2\cosθ) \bigr] dθ$ 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 $2α\ge |β|$ and $\eps>0$, then the corresponding matrix is Szeg\H o.