Magnetic Perturbations of Anyonic and Aharonov-Bohm Schr\"{o}dinger Operators

TL;DR

This paper investigates the self-adjoint extensions of Schrödinger operators for anyons and particles with Aharonov-Bohm effects in magnetic fields, providing a comprehensive classification for certain potentials.

Contribution

It introduces a one-parameter family of self-adjoint realizations for the Hamiltonian of anyons and classifies all extensions for specific magnetic potentials.

Findings

Identified a one-parameter family of self-adjoint operators

Classified all extensions for certain magnetic potentials

Analyzed models with Aharonov-Bohm singularities

Abstract

We study the Hamiltonian describing two anyons moving in a plane in presence of an external magnetic field and identify a one-parameter family of self-adjoint realizations of the corresponding Schr\"{o}dinger operator. We also discuss the associated model describing a quantum particle immersed in a magnetic field with a local Aharonov-Bohm singularity. For a special class of magnetic potentials, we provide a complete classification of all possible self-adjoint extensions.