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Young Seminar
Organizers: Douglas Lundholm (IHP) & Julien Sabin (Cergy)


Jean-Claude Cuenin (Imperial College): "Block-Diagonalization of Operators with Gaps, with Applications to Dirac Operators"
Wednesday 22 May 2013, 14:00 - 15:00
Salle 05


Abstract: It is well known that the spectrum of the Dirac operator is not bounded from below. This presents one of the major challenges for a consistent one-particle interpretation of the Dirac theory. In the field-free case the Foldy-Wouthuysen transformation exactly decouples the positive and negative spectrum. In the presence of external fields one has to resort to less direct methods. H. Siedentop and E. Stockmeyer have demonstrated the existence of an exact transformation for the Dirac operator with a Coulomb potential $V=-Z alpha/|cdot|$ (where $alpha$ is the fine structure constant) up to nuclear charge $Z=93$. Moreover, they showed that the formal Taylor series of the transformed operator in the coupling constant $gamma=Z alpha$ converges in the norm-resolvent sense up to $Z=51$, thus putting the so-called Douglas-Kroll-Hess method, widely used in quantum chemistry, on solid mathematical ground. In this talk, we will show how the above results can be generalized to potentials with stronger Coulomb singularities (up to $Z=124$ and $Z=62$, respectively). The method is based on the indefinite inner product space approach initiated by H. Langer and C. Tretter and is general enough to be applicable to perturbations (in the quadratic form sense) of self-adjoint operators with a spectral gap at the origin.