On the continuous analog of Rakhmanovʹs Theorem for orthogonal polynomials

We obtain the continuous analogs of Rakhmanovʹs 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.