On the continuous analog of Rakhmanov's Theorem for orthogonal polynomials

We obtain the continuous analogs of Rakhmanov's Theorem for polynomials orthogonal on the unit circle. Sturm-Liouville operators and Krein systems are considered. For a Sturm-Liouville operator with bounded potential q, we prove the following statement. If the essential spectrum and absolutely continuous component of the spectral measure fill the whole positive half-line, then q decays at infinity in the certain integral sense.