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 constantL 0,3 which is far worse than the best available by Lieb (for scalar potentials). However, it leads to a uniform bound (in the dimensiond≥3) for the quotientL 0,d/Lcl 0,d is the so-called classical constant. This gives some improvement in large dimensions.
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Hundertmark, D. On the number of bound states for Schrödinger operators with operator-valued potentials. Ark. Mat. 40, 73–87 (2002). https://doi.org/10.1007/BF02384503
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DOI: https://doi.org/10.1007/BF02384503