Proper waves of a weakly guiding waveguide in a half-space

The problem for eigenwaves of weakly guiding dielectric waveguide in the half-space is reduced to nonlinear eigenvalue problem for holomorphic Fredholm operator-valued function. The problem for surface waves is reduced to the linear eigenvalue problem for integral operator with symmetric, positive, weakly polar kernel. The existence, localization, and dependence on parameters of the spectrum are investigated.