The single-particle density of states and a resonance in the Aharonov-Bohm potential

TL;DR

This paper investigates the density of states and resonances in the Aharonov-Bohm potential, revealing a link between bound states and resonances, and exploring implications for electron propagation and Hall effect.

Contribution

It introduces a detailed analysis of the DOS and resonance phenomena in the Aharonov-Bohm potential, including the application of the Krein-Friedel formula with zeta function regularization.

Findings

Bound states are always accompanied by non-Breit-Wigner resonances.

The differential scattering cross section becomes asymmetric with bound states.

Hall resistivity is derived from electron propagation in vortex-like potentials.

Abstract

The single-particle densitity of states (DOS) for the Pauli and the Schr\"{o}dinger Hamiltonians in the presence of an Aharonov-Bohm potential is calculated for different values of the particle magnetic moment. The DOS is a symmetric and periodic function of the flux. The Krein-Friedel formula can be applied to this long-ranged potential when regularized with the zeta function. We have found that whenever a bound state is present in the spectrum it is always accompanied by a resonance. The shape of the resonance is not of the Breit-Wigner type. The differential scattering cross section is asymmetric if a bound state is present and gives rise to the Hall effect. As an application, propagation of electrons in a dilute vortex limit is considered and the Hall resistivity is calculated.