Quadratic forms for Aharonov-Bohm Hamiltonians

TL;DR

This paper uses quadratic form methods to analyze various self-adjoint realizations of a 2D Schrödinger operator with Aharonov-Bohm flux and perturbations, including flux limit behavior.

Contribution

It introduces a quadratic form framework to characterize self-adjoint extensions of Aharonov-Bohm Hamiltonians with singular perturbations.

Findings

Characterization of self-adjoint realizations via quadratic forms

Analysis of the Friedrichs Hamiltonian and singular perturbations

Limit behavior of the Friedrichs Hamiltonian as flux approaches zero

Abstract

We consider a charged quantum particle immersed in an axial magnetic field, comprising a local Aharonov-Bohm singularity and a regular perturbation. Quadratic form techniques are used to characterize different self-adjoint realizations of the reduced two-dimensional Schr\"odinger operator, including the Friedrichs Hamiltonian and a family of singular perturbations indexed by $2 \times 2$ Hermitian matrices. The limit of the Friedrichs Hamiltonian when the Aharonov-Bohm flux parameter goes to zero is discussed in terms of $\Gamma$ - convergence.