Abstract: We study the spectrum of a family $A_\alpha$ of self-adjoint partial differential operators, depending on a real parameter $\alpha$. The differential expression which defines the action of the operator does not involve the parameter, it appears only in the boundary conditions. From the point of view of the Perturbation Theory, we are dealing with the operators, defined via their quadratic forms, and the perturbation is only form-bounded, but not form-compact with respect to the unperturbed operator. This situation is rather unusual for this class of problems. This is reflected in the character of results. There exists a "borderline" value $\alpha_0$, such that the spectral properties of $A_\alpha$ for $\alpha<\alpha_0$ and for $\alpha>\alpha_0$ are quite different which can be interpreted as phase transition. To a large extent, these properties are determined by an auxiliary Jacobi matrix, which also depends on $\alpha$. The family studied was suggested by U. Smilansky as a model of an irreversible quantum system. A large part of the results was obtained in cooperation with S.N. Naboko.