Mathematical Physics

Bernd Thaller

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 [7], [9], [1].
  • A supersymmetric proof that singularities produced by anomalous magnetic moments have in fact a regularizing influence in the Dirac equation [11].
  • 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 [13].
  • 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 [14], [15], [17].
  • 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) [20].

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