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 similar
to an integral operator with a logarithmic kernel depending on the sum of arguments. By
a simple change of variables, the corresponding equation is reduced to an integral equation of
convolution type defined on a finite interval (as is well known, such equations in the general case
cannot be solved by quadratures). Next, using the Fourier transform, the equation is reduced to
a conjugation problem for analytic functions and then to an infinite system of linear algebraic
equations, the isolation of the main terms in which allows deriving a relation that determines
the spectrum of the original problem.