Universal estimate of the gap for the Kac operator in the convex case |
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Authors: | Bernard Helffer |
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Institution: | 1. URA CNRS 762 DMI-ENS, 45, rue d'Ulm, F-75230, Paris Cédex, France
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Abstract: | The aim of this paper is to prove that ifV is a strictly convex potential with quadratic behavior at ∞, then the quotient μ2/μ1 between the largest eigenvalue and the second eigenvalue of the Kac operator defined on L2(? m ) by exp ?V(x)/2 · exp Δx · exp ?V(x)/2 where Δx is the Laplacian on ? m satisfies the condition: $${{\mu _2 } \mathord{\left/ {\vphantom {{\mu _2 } {\mu _1 {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} \mathord{\left/ {\vphantom {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} {2,}}} \right. \kern-\nulldelimiterspace} {2,}}}}} \right. \kern-\nulldelimiterspace} {\mu _1 {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} \mathord{\left/ {\vphantom {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} {2,}}} \right. \kern-\nulldelimiterspace} {2,}}}}$$ where σ is such that HessV(x)≥σ>0. |
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