Solution of a Homogeneous Version of Love Type Integral Equation in Different Asymptotic Regimes

We consider one-dimensional convolution integral equation on an interval of Fredholm second kind whose particular non-homogeneous versions are known as Love, Gaudin and Lieb-Liniger equation. From operator-theoretic point of view, we are facing a problem of spectral decomposition of a compact integral operator that is finding its eigenvalues and eigenfunctions. We provide methods of deducing the eigenvalues and eigenfunctions in two asymptotic regimes depending on the size of the interval. In the case of small interval, the problem is essentially approximated by another one whose solutions are close to prolate spheroidal wave functions (Slepian functions). In the case of large interval, the problem is reduced to an auxiliary integro-differential equation which is treated by Wiener-Hopf type technique. We illustrate the obtained asymptotical results in both cases by comparing them with direct numerical solution of the integral equation by collocation method. It is remarkable that even though solutions are close to trigonometric functions, they are not exactly equal to them. This fact is in contrast with the results of known constructive approaches to homogeneous Fredholm equations of second kind.