On the Asymptotic Behavior of Eigenvalues and Eigenfunctions of an Integral Convolution Operator with a Logarithmic Kernel on a Finite Interval

Abstract

It is well known that one-dimensional integral equations of the convolution type considered on a finite interval cannot generally be solved by quadratures in contrast to similar equations considered on the entire line or on the half-line. For this reason, when studying their spectrum, some asymptotic method must be used. The paper considers a convolution-type integral operator with a logarithmic kernel defined on a finite interval. Using the Fourier transform, the problem is successively reduced to a conjugation problem and to a singular integral equation on the half-line in which the integral operator is not a contraction. It is shown that the leading part of the integral equation thus obtained can be inverted explicitly. The cases of even and odd eigenfunctions are considered, in each of which the asymptotics of the eigenvalues and eigenfunctions of the original operator is found.