1112.0673 (Laszlo Erdos et al.)

Relativistic Scott correction in self-generated magnetic fields    [PDF]

Laszlo Erdos, Soren Fournais, Jan Philip Solovej

We consider a large neutral molecule with total nuclear charge $Z$ in a model
with self-generated classical magnetic field and where the kinetic energy of
the electrons is treated relativistically. To ensure stability, we assume that
$Z \alpha < 2/\pi$, where $\alpha$ denotes the fine structure constant. We are
interested in the ground state energy in the simultaneous limit $Z \rightarrow
\infty$, $\alpha \rightarrow 0$ such that $\kappa=Z \alpha$ is fixed. The
leading term in the energy asymptotics is independent of $\kappa$, it is given
by the Thomas-Fermi energy of order $Z^{7/3}$ and it is unchanged by including
the self-generated magnetic field. We prove the first correction term to this
energy, the so-called Scott correction of the form $S(\alpha Z) Z^2$. The
current paper extends the result of \cite{SSS} on the Scott correction for
relativistic molecules to include a self-generated magnetic field. Furthermore,
we show that the corresponding Scott correction function $S$, first identified
in \cite{SSS}, is unchanged by including a magnetic field. We also prove new
Lieb-Thirring inequalities for the relativistic kinetic energy with magnetic
fields.

View original: http://arxiv.org/abs/1112.0673