On General Cwikel–Lieb–Rozenblum and Lieb–Thirring Inequalities

Part of the International Mathematical Series book series (IMAT, volume 13)


These classical inequalities allow one to estimate the number of negative eigenvalues and the sums \(S_\gamma = \sum { |\lambda _i |^\gamma } \) for a wide class of Schrödinger operators. We provide a detailed proof of these inequalities for operators on functions in metric spaces using the classical Lieb approach based on the Kac–Feynman formula. The main goal of the paper is a new set of examples which include perturbations of the Anderson operator, operators on free, nilpotent, and solvable groups, operators on quantum graphs, Markov processes with independent increments. The study of the examples requires an exact estimate of the kernel of the corresponding parabolic semigroup on the diagonal. In some cases the kernel decays exponentially as t → ∞. This allows us to consider very slow decaying potentials and obtain some results that are precise in the logarithmical scale.


Markov Process Symmetry Condition Heisenberg Group Erential Operator Negative Eigenvalue 
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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of North Carolina at CharlotteCharlotteUSA

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