Stability and semiclassics in self-generated fields

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 $β\int B^2$ 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 $β$ 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 $E(β, h, V)$. The relevant parameter measuring the field strength in the semiclassical limit is $κ=βh$. We are not able to give the exact leading order semiclassical asymptotics uniformly in $κ$ or even for fixed $κ$. We do however give upper and lower bounds on $E$ with almost matching dependence on $κ$. In the simultaneous limit $h\to0$ and $κ\to\infty$ 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.

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