Results on the Aharonov-Bohm effect without contact with the solenoid

TL;DR

This paper rigorously verifies the Aharonov-Bohm effect by analyzing a modified quantum model with a confining potential, showing the spectrum's dependence on magnetic flux without contact with the solenoid.

Contribution

It introduces a new self-adjoint extension of the Hamiltonian with a confining potential, demonstrating the flux-dependent spectral properties.

Findings

Spectrum is discrete with a nonconstant 1-periodic eigenvalue function.

Eigenvalue minima occur at integer fluxes, maxima at half-integers.

The model confirms the Aharonov-Bohm effect without particle contact with the solenoid.

Abstract

We add a confining potential to the Aharonov-Bohm model resulting in no contact of the particle with the solenoid (border); this is characterized by a unique self-adjoint extension of the initial Hamiltonian operator. It is shown that the spectrum of such extension is discrete and the first eigenvalue is found to be a nonconstant 1-periodic function of the magnetic flux circulation with a minimum at integers and maximum at half-integer circulations. This is a rigorous verification of the effect.