S.A. Denisov
On the existence of the absolutely continuous component for the spectral
measure associated with some Krein systems and Sturm-Liouville operators.
(36K, LATeX)
ABSTRACT. We consider 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, we prove that the essential support of the
absolutely continuous component of the spectral measure is
$[m,\infty)$ if $\limsup_{x\to\infty} q(x)=m$ and
$q^{\prime}\in L^2(R^+)$. That holds for arbitrary conditions
at zero. This result partially solves one open problem stated
recently by S.Molchanov, M.Novitskii, and B.Vainberg.