Self-adjoint extension approach for singular Hamiltonians in (2+1) dimensions

TL;DR

This paper reviews two methods based on self-adjoint extensions to analyze singular Hamiltonians in (2+1) dimensions, focusing on boundary conditions and physical states in quantum systems with topological defects.

Contribution

It compares two self-adjoint extension methods for singular Hamiltonians and applies them to a spin 1/2 particle in Aharonov-Bohm potential within conical space.

Findings

Derived boundary conditions for singular Hamiltonians.

Analyzed bound and scattering states in the Aharonov-Bohm setup.

Showed the applicability of methods to topological defect models.

Abstract

In this work, we review two methods used to approach singular Hamiltonians in (2+1) dimensions. Both methods are based on the self-adjoint extension approach. It is very common to find singular Hamiltonians in quantum mechanics, especially in quantum systems in the presence of topological defects, which are usually modelled by point interactions. In general, it is possible to apply some kind of regularization procedure, as the vanishing of the wave function at the location of the singularity, ensuring that the wave function is square-integrable and then can be associated with a physical state. However, a study based on the self-adjoint extension approach can lead to more general boundary conditions that still gives acceptable physical states. We exemplify the methods by exploring the bound and scattering scenarios of a spin 1/2 charged particle with an anomalous magnetic moment in the Aharonov-Bohm potential in the conical space.