Master Thesis Presentation - Dimitrios Kosmidis

Title: Essentially Self-Adjointness Of Singular Magnetic Laplace Beltrami Operator On Singular Riemmanian Manifolds

Abstract:

Essential self-adjointness is a fundamental global property of a differential operator which guarantees that the quantum system has a unique time evolution without requiring boundary conditions. Physically this is denoted as quantum confinement, and it is still an active area of research. This thesis examines the quantum geometric magnetic confinement problem on a singular Riemannian manifold. Specifically, we analyze the conditions required to achieve essential self-adjointness for a magnetic Laplace-Beltrami operator in the presence of singular metric and magnetic boundaries where the corresponding geometric quantities are degenerate.

To determine the criteria for confinement, we examine two distinct cases: first, when the metric and magnetic boundaries are separated by a strictly positive distance, and second, when the two boundaries completely coincide. Our analysis depends on a major assumption by transforming the magnetic potential into a divergence-free form using the Hodge-Helmholtz decomposition theorem. Lower bounds on the spectral norm of the magnetic field singularity alongside the effective potentials generated by the metric smooth measure, establish criteria for confinement. These criteria define sufficient bounds where the singular magnetic field or the metric dominates the system, ensuring that quantum particle are confined within the manifold.

Supervisors: Prof. Marcello Serie, Prof. Holgerwaalkens