Spectral characteristics of some nonlocal boundary-value problems

We consider two problems on eigenvalues of a nonlocal boundary-value problem for Laplace operator over a two-dimensional disk. We write out the adjoint boundary-value problems and show these problems have only eigenfunctions, but no associated functions. We also show that the spectrum of these problems does not lie in the Carleman parabola and the system of eigenfunctions, although complete and minimal in L2 is not a basis in L2. It is proved that, under certain assumptions a given function can be expanded into a biorthogonal series in the eigenfunctions of these problems, the series is uniformly convergent.