On the number of bound states for Schrodinger operators with operator-valued potentials

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 Schrodinger operators with operator-valued potentials. We recover Cwikel's bound for the Lieb-Thirring constant L-0,L-3 which is far worse than the best available by Lieb (for scalar potentials). However, it leads to a uniform bound (in the dimension dgreater than or equal to3) for the quotient L-0,L-d/L-0,d(cl), where L-0,d(cl) is the so-called classical constant. This gives some improvement in large dimensions.