Spectral analysis of relativistic atoms -- Dirac operators with singular potentials (Q845262)

This is the first part of a series of two papers, which investigate spectral properties of Dirac operators with singular potentials and show that the lifetime of excited states of a relativistic one-electron atom obeys Fermi's Golden Rule and coincides with the non-relativistic result in leading order in the fine structure constant. The main technical tool is complex dilation in connection with the Feshbach projection method. In this paper, the author generalizes the spectral analysis of \textit{P. Šeba} [Lett. Math. Phys. 16, No. 1, 51--59 (1988; Zbl 0659.47017)] and \textit{R. A. Weder} [Ann. Soc. Sci. Bruxelles, Sér. I 87, 341--355 (1973)] to operators with Coulomb type potentials, which are not relatively compact perturbations. In particular, the necessary properties of complex dilated spectral projections are derived and the non-relativistic limit of complex dilated Dirac operators is discussed. The results in this paper are applied in the second part [\textit{M. Huber}, Doc. Math., J. DMV 14, 115--156 (2009; Zbl 1173.81026)] to investigate resonances in a relativistic Pauli-Fierz model.