01-236 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) Jul 2, 01

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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.

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