Department of Mathematics - Mathematics Colloquium - Differential Operators with Singular Potentials

Schrödinger and Dirac operators with singular interactions supported on curves and surfaces represent a mathematically interesting class of differential operators in which one is able to derive numerous relations between spectral and geometric properties. At the same time such Hamiltonians are physically useful, for instance in quantum mechanics, where they describe various nanostructures as, e.g., leaky quantum graphs or graphene, as well as in the theory of photonic crystals. This is one of the main reasons why such operators with delta-interactions supported on manifolds have attracted a lot of attention by mathematicians and physicists in the recent past. The aim of this lecture is to give an introduction to the topic and to discuss some qualitative spectral properties of self-adjoint Schrödinger and Dirac operators with singular potentials. We first briefly review some classical results for regular potentials from the literature and turn to more recent developments afterwards. One of our main objectives is to investigate Dirac operators with delta-potentials supported on curves or hyperplanes, where it is necessary to distinguish the so-called non-critical and critical cases for the strength of the singular perturbation. In particular, it turns out that Dirac operators with singular potentials in the critical case have some unexpected spectral properties.