01-106 B. Helffer, T. Ramond
Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator. (1203K, gzipped postscript) Mar 21, 01
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Abstract. We continue the study started by the first author of the semiclassical Kac Operator. This kind of operator has been obtained for example by M. Kac as he was studying a 2D spin lattice by the so-called "transfer operator method". We are interested here in the thermodynamical limit $\Lambda(h)$ of the ground state energy of this operator. For Kac's spin model, $\Lambda(h)$ is the free energy per spin, and the semiclassical regime corresponds to the mean-field approximation. Under suitable assumptions, which are satisfied by many examples comming from statistical mechanics, we construct a formal asymptotic expansion for $\Lambda(h)$ in powers of $h$, from which we derive precise estimates. We work in the setting of \emph{standard functions} introduced by J. Sj\"ostrand for the study of similar questions in the case of Schr\"odinger operators.

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