  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.
