The Pauli-Poisson equation and its semiclassical limit

Abstract: The Pauli-Poisson equation is a semi-relativistic model for charged spin-1/2-particles in a strong external magnetic field and a self-consistent electric potential computed from the Poisson equation in 3 space dimensions. It is a system of two magnetic Schr\"odinger equations for the two components of the Pauli 2-spinor, representing the two spin states of a fermion, coupled by the additional Stern-Gerlach term representing the interaction of magnetic field and spin. We study the global wellposedness in the energy space and the semiclassical limit of the Pauli-Poisson to the magnetic Vlasov-Poisson equation with Lorentz force and the semiclassical limit of the linear Pauli equation to the magnetic Vlasov equation with Lorentz force. We use Wigner transforms and a density matrix formulation for mixed states, extending the work of P. L. Lions & T. Paul as well as P. Markowich & N.J. Mauser on the semiclassical limit of the non-relativistic Schr\"odinger-Poisson equation.