The most general admissible boundary conditions are derived for an idealized AharonovBohm flux intersecting the plane at the origin on the background of a homogeneous magnetic field. A standard technique based on self-adjoint extensions yields a four-parameter family of boundary conditions; the other two parameters of the model are the AharonovBohm flux and the homogeneous magnetic field. The generalized boundary conditions may be regarded as a combination of the AharonovBohm effect with a point interaction. Spectral properties of the derived Hamtonians are studied in detail.
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 propose a model for scattering in a flat resonator with a thin antenna. The results are applied to rectangular microwave cavities. We compute the resonance spacing distribution and show that it agrees well with experimental data provided the antenna radius is much smaller than wavelengths of the resonance wavefunctions.
In addition to the conventional renormalized-coupling-constant picture, point interactions in two and three dimensions are shown to model within a suitable energy range scattering on localized potentials, both attractive and repulsive.