Laszlo Erdos, Jan Philip Solovej
Uniform Lieb-Thirring inequality for the three dimensional
Pauli operator with a strong non-homogeneous magnetic field
(211K, latex)
ABSTRACT. 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.