Spectrum of the Riemann–Hilbert–Poincaré problem for analytic functions

We study the Riemann–Hilbert–Poincaré boundary value problem for analytic function. This problem will lead to inhomogeneous Fuchsian differential equations. We find that its spectrum is not characterized by the smoothness of its coefficient on the boundary but by its interior analytic continuation property. Moreover, the multiplicities of eigenfunctions for different eigenvalues are not necessarily the same even when the eigenvalues are small.