Single-Particle Density of States for the Aharonov-Bohm Potential and Instability of Matter with Anomalous Magnetic Moment in 2+1 Dimensions

TL;DR

This paper investigates the density of states and stability of matter in 2+1 dimensions under the Aharonov-Bohm potential, revealing conditions for instability related to magnetic moment and the presence of bound states.

Contribution

It demonstrates the instability of matter with magnetic moments greater than 2 in the presence of Aharonov-Bohm flux, extending results to the Klein-Gordon equation with Pauli coupling.

Findings

Matter becomes unstable when magnetic moment exceeds 2 with flux present.

Bound states are always accompanied by resonances proportional to their binding energy.

The Krein-Friedel formula is valid with zeta function regularization for this potential.

Abstract

In the nonrelativistic case we find that whenever the relation $mc^2/e^2 <X(\al,g_m)$ is satisfied, where $\al$ is a flux in the units of the flux quantum, $g_m$ is magnetic moment, and $X(\al,g_m)$ is some function that is nonzero only for $g_m>2$ (note that $g_m=2.00232$ for the electron), then the matter is unstable against formation of the flux $\al$. The result persists down to $g_m=2$ provided the Aharonov-Bohm potential is supplemented with a short range attractive potential. We also show that whenever a bound state is present in the spectrum it is always accompanied by a resonance with the energy proportional to the absolute value of the binding energy. is considered. For the Klein-Gordon equation with the Pauli coupling which exists in (2+1) dimensions without any reference to a spin the matter is again unstable for $g_m>2$. The results are obtained by calculating the change of the density of states induced by the Aharonov-Bohm potential. The Krein-Friedel formula for this long-ranged potential is shown to be valid when supplemented with zeta function regularization. PACS : 03.65.Bz, 03-70.+k, 03-80.+r, 05.30.Fk