Scattering by an Ultralocal Potential in a Non-trivial Topology

Scattering of non-relativistic particles by an ultralocal (δ-) potential is considered in two-dimensional manifolds with various topology (cylinder, torus, sphere, and Lobachevski plane). The behavior of the bound state energy as a function of the geometrical and topological characteristics of the space is studied. It is shown that for the compact non-simply connected manifolds of small radius the variation of the twisting angles (Aharonov-Bohm fluxes) may lead to delocalization of the bound state. For a simply connected geometry the influence of curvature on the bound state is considered and the possibility of "geometric delocalization" of the impurity levels is demonstrated explicitly for the spaces of constant curvature. We also consider the Aharonov-Bohm effect for the anyons on a cylinder. It is shown that a local regular potential can induce the Aharonov-Bohm oscillations in the anyon gas with anomalous (non-mesoscopic) dependence of oscillation amplitude on the geometrical sizes of the system.