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.

abstract

In this paper, a class of FETI-DP preconditioners is developed for a fast solution of the linear system arising from staggered discontinuous Galerkin discretization of the two-dimensional Stokes equations. The discretization has been recently developed and has the distinctive advantages that it is optimally convergent and has a good local conservation property. In order to efficiently solve the linear system, two kinds of FETI-DP preconditioners, namely, lumped and Dirichlet preconditioners, are considered and analyzed. Scalable bounds C (H/h) and C (1+ log (H/h)) 2 are proved for the lumped and Dirichlet preconditioners, respectively, with the constant C depending on the infsup constant of the discrete spaces but independent of any mesh parameters. Here H/h stands for the number of elements across each subdomain. Numerical results are presented to confirm the theoretical estimates.

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.