Foundations of Electro-Magnetic Laplacian Under Low-Regularity Potentials

Abstract

This thesis aims to establish a rigorous mathematical framework for the electromagnetic Laplacian, in the context of low-regularity vector potential and scalar potential. While the investigation of the formal self-adjointness of this unbounded operator is straightforward, establishing its self-adjointness is a fundamental step in exploring its spectral theory. The work focuses on proving self-adjointness using the theories of quadratic forms (Friedrichs' Theorem) and perturbation theory (Kato–Rellich Theorem). It further investigates the spectral properties of the operator, particularly its essential spectrum, and examines the specific cases of the magnetic harmonic oscillator and Landau Hamiltonian. The thesis also provides a self-contained presentation of known results currently scattered throughout the literature.