The Maxwell-Pauli Equations

Abstract

We study the quantum mechanical many-body problem of $N\ge 1$ non-relativistic electrons with spin interacting with their self-generated classical electromagnetic field and $K\ge 0$ static nuclei. We model the dynamics of the electrons and their self-generated electromagnetic field using the so-called many-body Maxwell-Pauli equations. The main result of this thesis is to construct time global, finite-energy, weak solutions to the many-body Maxwell-Pauli equations under the assumption that the fine structure constant $\alpha$ and the nuclear charges are not too large. The assumptions on the size of $\alpha$ and the nuclear charges ensure that we have energetic stability for this system, i.e., the absolute ground state energy exists. The work in this thesis serves as an initial step towards understanding the connection between the energetic stability of matter in quantum mechanics and the well-posedness of the corresponding dynamical equations.

arXiv comment95 pages. PhD thesis, Georgia Institute of Technology, May 2020. Includes results from arXiv:1904.06748

References in corpus (10)

• Stability of Matter in Magnetic Fields
• Self-adjointness of Dirac operators via Hardy-Dirac inequalities
• Stability and Instability of Relativistic Electrons in Classical Electro magnetic Fields
• Stability of a Model of Relativistic Quantum Electrodynamics
• Global Existence and Uniqueness of Solutions to the Maxwell-Schr{ö}dinger Equations
• Long Range Scattering and Modified Wave Operators for the Maxwell-Schr"odinger System II. The general case
• Global wellposedness in the energy space for the Maxwell-Schrödinger system
• Existence of a Unique Local Solution to the Many-body Maxwell-Schrödinger Initial Value Problem
• Nonlinear Maxwell-Schroedinger system and Quantum Magneto-Hydrodynamics in 3D
• Existence of Travelling Wave Solutions to the Maxwell-Pauli and Maxwell-Schrödinger Systems