In this note we prove that the essential spectrum of a Schrdinger operator with potential supported on a finite number of compact Lipschitz hypersurfaces is given by [0, +). We emphasize that the union of a family of Lipschitz hypersurfaces is in general not Lipschitz. ( 2013 WileyVCH Verlag GmbH & Co. KGaA, Weinheim)

We discuss a modification of Smilansky model in which a singular potential channel is replaced by a regular, below unbounded potential which shrinks as it becomes deeper. We demonstrate that, similarly to the original model, such a system exhibits a spectral transition with respect to the coupling constant, and determine the critical value above which a new spectral branch opens. The result is generalized to situations with multiple potential channels.

We study a two-dimensional charged particle interacting with a magnetic field, in general non-homogeneous, perpendicular to the plane, a confining potential, and a point interaction. If the latter moves adiabatically along a loop, the state corresponding to an isolated eigenvalue acquires a Berry phase. We derive an expression for it and evaluate it in several examples such as a homogeneous field, a magnetic whisker, a particle confined at a ring or in quantum dots, a parabolic and a zero-range one. We also discuss the behavior of the lowest Landau level in this setting obtaining an explicit example of the WilczekZee phase for an infinitely degenerated eigenvalue.

Quantum networks are often modelled using Schrdinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article we give a survey, technically not too heavy, of several recent results which serve this purpose. Specifically, we consider approximations by means of fat graphsin other words, suitable families of shrinking manifoldsand discuss convergence of the spectra and resonances in such a setting.

We treat a free spinless quantum particle moving on a configuration manifold which consists of two identical parts connected in one point. Most attention is paid to the three-dimensional case when these parts are halfspaces with Neumann condition on the boundary; we also discuss briefly a more general boundary conditions. The class of admissible Hamiltonians is constructed by means of the theory of self-adjoint extensions. Among them, particularly important is a two-parameter family whose elements are invariant with respect to exchange of the halfspaces; we compute the transmission coefficient for each of these extensions. We discuss also the motion on two planes considered in our recent paper, obtaining another characterization of the admissible Hamiltonians. In conclusion, the two situations are compared as models for point-contact spectroscopical experiments in thin metal films.