01-33 Dirk Hundertmark

On the number of bound states for Schr\"odinger operators with operator-valued potentials (41K, LaTeX2e) Jan 22, 01

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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.

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