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

Abstract

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.