Here is a selection of my scientific achievements in the field of Mathematical Physics (numbers refer to the publication list). Much of this work is centered around the Dirac equation.
- Derivation of a formula for the first and second-order relativistic corrections of eigenvalues of Schrödinger operators. First mathematically rigorous proof of the Foldy-Wouthuysen result (by analytic perturbation theory) about thirty years after its introduction , , .
- A supersymmetric proof that singularities produced by anomalous magnetic moments have in fact a regularizing influence in the Dirac equation .
- Established the asymptotic completeness of the Dirac equation with Coulomb potentials (using geometric methods and asymptotic observables). Asymptotic completeness is the basic mathematical requirement for the existence of a scattering theory .
- First proof of asymptotic completeness for the Schrödinger equation with long-range magnetic fields and for the Dirac equation with long-range potentials of a more general (non-electrostatic) type. Discovery of the fact that a certain class of long-range magnetic fields are in fact short-range in the Poincaré gauge , , .
- Proof of the first comparison theorem for Schrödinger operators with arbitrary magnetic fields in two dimensions (sort of diamagnetic inequality in the form of a heat-kernel estimate) .
I have made contributions to the following research areas:
- essential self-adjointness
- scattering theory by stationary and time-dependent methods
- asymptotic completeness for the Dirac eqation with long-range potentials
(also in the case of non-electrostatic interactions, in particular magnetic fields)
- the nonrelativistic limit of the Dirac equation
- supersymmetry and normal forms of the Dirac equation
- heat kernel estimates for the Schrödinger equation with magnetic fields
- degenerate Cauchy problems