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

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 Schrödinger operators with operator-valued potentials. We recover Cwikel's bound for the Lieb-Thirring constant L0,3 which is far worse than the best available by Lieb (for scalar potentials). However, it leads to a uniform bound (in the dimension d≥3) for the quotient L0,d/L0,dcl, where L0,dcl is the so-called classical constant. This gives some improvement in large dimensions.