Journal of the European Mathematical Society
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Stability and semiclassics in self-generated fieldsLászló Erdős, Søren Fournais and Jan Philip Solovej (1) Institute of Scienceand Technology Austria, Am Campus 1, 3400, Klosterneuburg, Austria
(2) Department of Mathematics, University of Aarhus, Ny Munkegade 118, 8000, Aarhus C, Denmark
(3) Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100, 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