B. Helffer, T. Ramond
Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator.
(1203K, gzipped postscript)
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.