Spectral analysis of one class of the integro-differential operators

Abstract

Let ℋ = L2[0,ω] be the Hilbert space of equivalence classes of square integrable Lebesgue measurable complex functions on the closed interval [0,ω]. In this paper, we study the spectral properties of a second-order differential operator which perturbed by an integral operator. The integral operator belong to the two-sided ideal of Hilbert-Schmidt operator in ℋ. We use similar operators method for studying this operator. The similar operators method was developed by A. G. Baskakov and his collaborators. This method allows us to reduce the study of the operator to the one with a diagonal and a block-diagonal matrix. Asymptotic estimates of eigenvalues, eigenvectors, and spectral projections of an integro-differential operator are obtained.