Asymptotic behaviour of the spectra of integral convolution operators on a finite interval with homogeneous polar kernels

We obtain asymptotic formulae for the eigenvalues of integral convolution operators on a finite interval with homogeneous polar (complex) kernels. In the Fourier–Laplace images, the eigenvalue and eigenfunction problems are reduced to the Hilbert linear conjugation problem for a holomorphic vector-valued function with two components. This problem is in turn reduced to a system of integral equations on the half-line, and analytic properties of solutions of this system are studied in the Mellin images in Banach spaces of holomorphic functions with fixed poles. We study the structure of the canonical matrix of solutions of this Hilbert problem at the singular points, along with its asymptotic behaviour for large values of the reduced spectral parameter. The investigation of the resulting characteristic equations yields three terms (four in the positive self-adjoint case) of the asymptotic expansions of the eigenvalues, along with estimates of the remainders.