The Pauli-Poisson equation and its semiclassical limit

• Published in: Communications in partial differential equations Vol. 50; no. 1-2; pp. 130 - 161
• Main Author: Möller, Jakob
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 01.02.2025; Taylor & Francis Ltd
• Subjects: Asymptotic Analysis; Fermions; Lorentz force; Magnetic fields; magnetic Schrödinger equation; Pauli equation; Poisson equation; Relativistic effects; Schrodinger equation; Vlasov equations; Vlasov-Poisson equation; Wigner measures
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2024.2439358

The Pauli-Poisson equation is a semi-relativistic model for charged spin-1∕2-par-ticles in a strong external magnetic field and a self-consistent electric potential computed from the Poisson equation in three space dimensions. It is a system of two magnetic Schrödinger 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 well-posedness 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 Schrödinger-Poisson equation.