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

We study a spectral problem in a bounded domain ${\Omega \subset \mathbb{R}^{m}}$, depending on a bounded operator coefficient $Q>0$ and a dissipation parameter $\alpha>0$. 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 $ L_2(\Omega)$. In model one- and two-dimensional problems we establish the localization of the eigenvalues and find critical values of $\alpha$.