Laszlo Erdos, Jan Philip Solovej
Magnetic Lieb-Thirring inequalities with optimal dependence 
on the field strength
(82K, LaTeX)

ABSTRACT.  The Pauli operator describes the energy of a nonrelativistic quantum 
particle with spin $\sfrac{1}{2}$ in a magnetic field and an external 
potential. Bounds on the sum of the negative eigenvalues are called 
magnetic Lieb-Thirring (MLT) inequalities. The purpose of this paper 
is twofold. First, we prove a new MLT inequality in a simple way. 
Second, we give a short summary of our recent proof of a more refined 
MLT inequality \cite{ES-IV} and we explain the differences between the 
two results and methods. The main feature of both estimates, compared 
to earlier results, is that in the large field regime they grow with 
the optimal (first) power of the strength of the magnetic field. As a 
byproduct of the method, we also obtain optimal upper bounds on the 
pointwise density of zero energy eigenfunctions of the Dirac operator.