Stability and semiclassics in self-generated magnetic fields

We consider non-interacting particles subject to an external electric potential $V$ and a self-generated magnetic field $B$. The total energy includes the magnetic field energy $\beta \int B^2$ and we minimize over all particle states and magnetic vector potentials (of finite field energy). We estimate the total ground state energy of the system in the semiclassical limit $h \rightarrow 0_{+}$. The relevant parameter measuring the field strength in the semiclassical limit is $\beta h$. We prove that when $\beta h \rightarrow +\infty$ the effect of the self-generated magnetic field is negligible in the sense that we recover the standard, non-magnetic Weyl asymptotics in this case.

This is joint work with L. Erd\”{o}s and J. P. Solovej.