We consider scattering of a three-dimensional particle on a finite family of potentials. For some parameter values the scattering wavefunctions exhibit nodal lines in the form of closed loops, which may touch but do not entangle. The corresponding probability current forms vortical singularities around these lines; if the scattered particle is charged, this gives rise to magnetic flux loops in the vicinity of the nodal lines. The conclusions extend to scattering on hard obstacles or smooth potentials.

Convergence theories and a deluxe dual and primal finite element tearing and interconnecting algorithm are developed for a hybrid staggered DG finite element approximation of H(curl) elliptic problems in two dimensions. In addition to the advantages of staggered DG methods, the basis functions of the new hybrid staggered DG method are all locally supported in the triangular elements, and a Lagrange multiplier approach is applied to enforce the global connections of these basis functions. The interface problem on the Lagrange multipliers is further reduced to the resulting problem on the subdomain interfaces, and a dual and primal finite element tearing and interconnecting algorithm with an enriched weight factor is then applied to the resulting problem. Our algorithm is shown to give a condition number bound of <i>C</i>(1+log(<i>H</i><i>h</i>))², independent of the two parameters, where <i>H</i><i>h</i> is the number of

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.

A BDDC (Balancing Domain Decomposition by Constraints) algorithm is developed and analyzed for a staggered discontinuous Galerkin (DG) finite element approximation of second order scalar elliptic problems. On a quite irregular subdomain partition, an optimal condition number bound is proved for two-dimensional problems. In addition, a sub-optimal but scalable condition number bound is obtained for three-dimensional problems. These bounds are shown to be independent of coefficient jumps in the subdomain partition. Numerical results are also included to show the performance of the algorithm.

We consider a Schrdinger particle on a graph consisting of <i>N</i> links joined at a single point. Each link supports a real locally integrable potential <i>V</i> _<i>j</i>; the self-adjointness is ensured by the type boundary condition at the vertex. If all the links are semi-infinite and ideally coupled, the potential decays as <i>x</i> ¹ along each of them, is nonrepulsive in the mean and weak enough, the corresponding Schrdinger operator has a single negative eigenvalue; we find its asymptotic behavior. We also derive a bound on the number of bound states and explain how the coupling constant may be interpreted in terms of a family of squeezed potentials.