A spectral Szego theorem on the real line

We characterize even measures $\mu =wdx+{\mu }_s$ on the real line with finite entropy integral ${\int }_R\frac{\log w(t)}{1+t^2}dt>-\mathrm{\infty }$ 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.