Erdos L., Solovej J.P.
Semiclassical eigenvalue estimates for the Pauli operator
with strong non-homogeneous magnetic fields: I. Non-asymptotic 
Lieb-Thirring type estimate 
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ABSTRACT.  We give the first Lieb-Thirring type estimate on the sum of the
negative eigenvalues of the Pauli operator that
behaves as the corresponding semiclassical expression
even in the case of strong non-homogeneous magnetic fields.
This enables us, in the companion paper \cite{ES-II},
to obtain the leading order semiclassical eigenvalue asymptotic,
which, in turn, leads to the proof of the validity of the
magnetic Thomas-Fermi theory of \cite{LSY-II}. Our
work generalizes the results of \cite{LSY-II} to non-homogeneous
magnetic fields.