Second order semiclassics with self-generated magnetic fields

TL;DR

This paper analyzes the semiclassical behavior of the Pauli operator with self-generated magnetic fields, showing that for smooth potentials the magnetic effects are negligible at leading order, while singular potentials exhibit non-vanishing subleading terms.

Contribution

It provides a rigorous semiclassical asymptotic analysis of the Pauli operator with self-generated magnetic fields, including error bounds and effects of Coulomb singularities.

Findings

Leading order energy matches non-magnetic Weyl term for smooth potentials.

Error bounds are smaller by a factor of h^{1+ε} for smooth potentials.

Non-vanishing subleading term for Coulomb singular potentials.

Abstract

We consider the semiclassical asymptotics of the sum of negative eigenvalues of the three-dimensional Pauli operator with an external potential and a self-generated magnetic field $B$. We also add the field energy $\beta \int B^2$ and we minimize over all magnetic fields. The parameter $\beta$ effectively determines the strength of the field. We consider the weak field regime with $\beta h^{2}\ge {const}>0$, where $h$ is the semiclassical parameter. For smooth potentials we prove that the semiclassical asymptotics of the total energy is given by the non-magnetic Weyl term to leading order with an error bound that is smaller by a factor $h^{1+\e}$, i.e. the subleading term vanishes. However, for potentials with a Coulomb singularity the subleading term does not vanish due to the non-semiclassical effect of the singularity. Combined with a multiscale technique, this refined estimate is used in the companion paper \cite{EFS3} to prove the second order Scott correction to the ground state energy of large atoms and molecules.