A spectral Szego theorem on the real line

TL;DR

This paper characterizes measures on the real line with finite entropy integral using inverse spectral theory and provides criteria for spectral measures of Krein strings to have converging logarithmic integrals.

Contribution

It introduces a spectral Szego theorem on the real line linking measures with finite entropy to Hamiltonians in inverse spectral theory.

Findings

Characterization of measures with finite entropy integral

Criterion for spectral measures of Krein strings

Connection between spectral measures and Hamiltonians

Abstract

We characterize even measures $\mu=wdx+\mu_s$ on the real line with finite entropy integral $\int_{R} \frac{\log w(t)}{1+t^2}dt>-\infty$ in terms of $2\times 2$ Hamiltonian generated by $\mu$ in the sense of inverse spectral theory. As a corollary, we obtain criterion for spectral measure of Krein string to have converging logarithmic integral.