On Asymptotics of the Spectrum of an Integral Operator with a Logarithmic Kernel of a Special Form

Abstract

We study the asymptotic behavior of the spectrum of an integral operator similarto an integral operator with a logarithmic kernel depending on the sum of arguments. Bya simple change of variables, the corresponding equation is reduced to an integral equation ofconvolution type defined on a finite interval (as is well known, such equations in the general casecannot be solved by quadratures). Next, using the Fourier transform, the equation is reduced toa conjugation problem for analytic functions and then to an infinite system of linear algebraicequations, the isolation of the main terms in which allows deriving a relation that determinesthe spectrum of the original problem.