Leaky quantum structures

The paper reviews spectral properties of a class of singular Schrödinger operators with the interaction supported by manifolds or complexes of codimension one, in particular, their relations to the geometric setting of the problem. We describe how they can be approximated by operators of other classes and how such approximations can be used. Furthermore, we present asymptotic expansions of the eigenvalues in terms of the parameters characterizing the coupling strength and geometric deformations. We also give an example illustrating the influence of a magnetic field of the Aharonov-Bohm type and describe briefly results about singular perturbation of Dirac operators.