On the Existence of the Absolutely Continuous Component for the Measure Associated¶with Some Orthogonal Systems

Abstract:

In this article, we consider two orthogonal systems: Sturm–Liouville operators and Krein systems. For Krein systems, we study the behavior of generalized polynomials at the infinity for spectral parameters in the upper half-plain. That makes it possible to establish the presence of absolutely continuous component of the associated measure. For Sturm–Liouville operator on the half-line with bounded potential q, we prove that essential support of absolutely continuous component of the spectral measure is [m,∞) if and q ^′∈L ²(R ⁺). That holds for all boundary conditions at zero. This result partially solves one open problem stated recently by S. Molchanov, M. Novitskii, and B. Vainberg. We consider also some other classes of potentials.