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# Spectral properties of a fourth-order differential operator with integrable coefficients

(Impact Factor: 0.3). 09/2010; 270(1):184-193. DOI: 10.1134/S0081543810030144

ABSTRACT

The aim of this paper is to study spectral properties of differential operators with integrable coefficients and a constant
weight function. We analyze the asymptotic behavior of solutions to a differential equation with integrable coefficients for
large values of the spectral parameter. To find the asymptotic behavior of solutions, we reduce the differential equation
to a Volterra integral equation. We also obtain asymptotic formulas for the eigenvalues of some boundary value problems related
to the differential operator under consideration.

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ABSTRACT: For the Sturm-Liouville boundary-value problem on a segment we construct asymptotics for , where are the eigenvalues, and for the normalized eigenfunctions of the form for any , where and are expressed explicitly in terms of the potential . Under the assumption that is a real summable function, the terms and are as .
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