  The Pauli operator describes the energy of a nonrelativistic quantum particle
with spin 1/2 in a magnetic field and an external potential. A new
Lieb-Thirring type inequality on the sum of the negative eigenvalues is
presented. The main feature compared to earlier results is that in the large
field regime the present estimate grows with the optimal (first) power of the
strength of the magnetic field. As a byproduct of the method, we also obtain an
optimal upper bound on the pointwise density of zero energy eigenfunctions of
the Dirac operator. The main technical tools are:
  (i) a new localization scheme for the square of the resolvent of a general
class of second order elliptic operators;
  (ii) a geometric construction of a Dirac operator with a constant magnetic
field that approximates the original Dirac operator in a tubular neighborhood
of a fixed field line. The errors may depend on the regularity of the magnetic
field but they are uniform in the field strength.
