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.