We sketch here two mathematical models intended to describe the point-contact spectroscopical experiments. A new item is added to the list of recently discovered applications of the self-adjoint extensions theory.

We consider an open quantum dot modeled by a straight hard-wall channel with a potential well. If this potential depends on the longitudinal variable only, the system exhibits embedded eigenvalues. They turn into resonances if the symmetry is violated, either by a magnetic field or by deformation of the well. We construct a perturbation theory of these resonances in the case of a weak perturbation and discuss other properties of the model.

No abstract uploaded!

We study a charged two-dimensional particle confined to a straight parabolic-potential channel and exposed to a homogeneous magnetic field under the influence of a potential perturbation W. If W is bounded and periodic along the channel, a perturbative argument yields the absolute continuity of the bottom of the spectrum. We show that it can have any finite number of open gaps provided that the confining potential is sufficiently strong. However, if W depends on the periodic variable only, we prove by the Thomas argument that the whole spectrum is absolutely continuous, irrespective of the size of the perturbation. On the other hand, if W is small and satisfies a weak localization condition in the the longitudinal direction, we prove by the Mourre method that a part of the absolutely continuous spectrum persists.

The distribution of property is established through various mechanisms. In this paper we study the acreage distribution of land plots owned by natural persons in the Zln Region of the Czech Republic. We show that the data are explained in terms of a simple model in which the inheritance and market behavior are combined.