Jan Philip Solovej: Semi-classics for Schrödinger and Pauli operators with self-generated magnetic fields

I will discuss the energy of fermions moving in a fixed exterior electric potential but with a magnetic field chosen so as to minimize the total energy when the magnetic field-energy is included. This can be considered as particles with self-generated magetic interactions. It amounts to minimizing the sum of the negative eigenvalues of a magnetic Schrödinger or Pauli operator when the magnetic field energy is included. In a semiclassical limit there are two natural parameters: The usual semiclassical parameter h and the coupling strength to the magnetic field. I will discuss the semiclassical limit in different parameter regimes. In certain regimes we have the standard Weyl law. For the Pauli operator, however, there are regimes where the asymptotics is not given by Weyl. We do not know the exact asymptotics, but have "almost" sharp bounds.