Spectral analysis of relativistic atoms -- Dirac operators with singular potentials

Abstract

This is the first part of a series of two papers, which investigate spectral properties of Dirac operators with singular potentials. We examine various properties of complex dilated Dirac operators. These operators arise in the investigation of resonances using the method of complex dilations. We generalize the spectral analysis of Weder citeWeder1973 and Šeba citeSeba1988 to operators with Coulomb type potentials, which are not relatively compact perturbations. Moreover, we define positive and negative spectral projections as well as transformation functions between different spectral subspaces and investigate the non-relativistic limit of these operators. We will apply these results in citeHuber2008O in the investigation of resonances in a relativistic Pauli-Fierz model, but they might also be of independent interest.