Dirk Hundertmark
On the number of bound states for Schr\"odinger operators 
with operator-valued potentials
(41K, LaTeX2e)

ABSTRACT.  Cwikel's bound is extended to an operator-valued setting. 
One application of this result is a semi-classical bound for 
the number of negative bound states for Schr\"odinger operators 
with operator-valued potentials. 
We recover Cwikel's bound for the Lieb--Thirring constant 
$L_{0,3}$ which is far worse than the best available by Lieb 
(for scalar potentials). However, it leads to a uniform bound 
(in the dimension $d\ge 3$) for the quotient 
$L_{0,d}/ L^{\text{cl}}_{0,d}$, where $L^{\text{cl}}_{0,d}$ is 
the so-called classical constant. This gives some improvement 
in large dimensions.