SODDIM - Schrödinger operators with delta and delta interactions on manifolds

Schrödinger operators with delta-interactions on curves and special hypersurfaces have been studied intensively in the last decades. Many of their analytic and spectral properties are well understood. In contrast to that, spectral problems for Schrödinger operators with delta-interactions on manifolds of higher co-dimension have been studied only very little. These more singular delta-interactions have applications in quantum many-body problems, where classical regular potentials are approximatively replaced by equivalent delta-potentials. In this project we propose a general rigorous definition for self-adjoint Schrödinger operators with delta-interactions on manifolds of arbitrary co-dimension. Our aim is to study the spectral properties of these operators and, in particular, we plan to focus on three explicit problems on the existence of bound states and their quantitative properties: The corner points problem, the many body problem, and the induced bound states problem.