Journal of the European Mathematical Society

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Volume 15, Issue 6, 2013, pp. 2093–2113
DOI: 10.4171/JEMS/416

Published online: 2013-10-16

Stability and semiclassics in self-generated fields

László Erdős[1], Søren Fournais[2] and Jan Philip Solovej[3]

(1) Institute of Scienceand Technology Austria, Klosterneuburg, Austria
(2) University of Aarhus, Denmark
(3) University of Copenhagen, Denmark

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 Schrödinger operator.

Keywords: semiclassical eigenvalue estimate, Maxwell-Pauli system, Scott correction

Erdős László, Fournais Søren, Solovej Jan Philip: Stability and semiclassics in self-generated fields. J. Eur. Math. Soc. 15 (2013), 2093-2113. doi: 10.4171/JEMS/416