On spectral properties of one boundary value problem with a surface energy dissipation

We study a spectral problem in a bounded domain Ω ⊂ Rm depending on a bounded operator coefficient Q > 0 and a dissipation parameter α > 0. In the general case we establish sufficient conditions ensuring that the problem has a discrete spectrum consisting of countably many isolated eigenvalues of finite multiplicity accumulating at infinity. We also establish the conditions, under which the system of root elements contains an Abel-Lidskii basis in the space L2(Ω). In model oneand two-dimensional problems we establish the localization of the eigenvalues and find critical values of α.