Stability and semiclassics in self-generated fields

TL;DR

This paper studies the stability and semiclassical behavior of non-interacting particles in a fixed potential coupled with a self-generated magnetic field, providing bounds on the ground state energy and establishing Weyl asymptotics in certain limits.

Contribution

It introduces bounds on the ground state energy for particles coupled with self-generated magnetic fields, analyzing the effects of field strength in the semiclassical limit.

Findings

Standard Weyl asymptotics hold when both the semiclassical parameter and field strength parameter tend to infinity.

Bounds on the total energy are established with nearly matching dependence on the field strength parameter.

Results apply to both spin-1/2 particles and spinless cases, showing the robustness of the asymptotics.

Abstract

We consider non-interacting particles subject to a fixed external potential $V$ and a self-generated magnetic field $B$. The total energy includes the field energy $\beta \int B^2$ and we minimize over all particle states and magnetic fields. In the case of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter $\beta$ tunes the coupling strength between the field and the particles and it effectively determines the strength of the field. We investigate the stability and the semiclassical asymptotics, $h\to0$, of the total ground state energy $E(\beta, h, V)$. The relevant parameter measuring the field strength in the semiclassical limit is $\kappa=\beta h$. We are not able to give the exact leading order semiclassical asymptotics uniformly in $\kappa$ or even for fixed $\kappa$. We do however give upper and lower bounds on $E$ with almost matching dependence on $\kappa$. In the simultaneous limit $h\to0$ and $\kappa\to\infty$ we show that the standard non-magnetic Weyl asymptotics holds. The same result also holds for the spinless case, i.e. where the Pauli operator is replaced by the Schr\"odinger operator.