The logarithmic integral and Mőller wave operators

I'm going to discuss a necessary and sufficient condition for the existence of wave operators of past and future for the unitary group generated by a one-dimensional Dirac operator on the positive half line. The criterion could be formulated both in terms of the operator potential and in terms of its spectral measure. In the second case, a necessary and sufficient condition for scattering coincides with the finiteness of the Szegő logarithmic integral
$$ \int_{\mathbb R} \frac{\log w}{1+x^2}dx > - \infty $$
of the density of the spectral measure. The proof essentially uses ideas from the theory of orthogonal polynomials on the unit circle, in particular, a formula discovered by S. Khrushchev.
Partially based on joint works with S. Denisov.