99-202 Helffer B., Martinez A.
Phase Transition in the Semiclassical Regime (58K, LATeX 2e) May 31, 99
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Abstract. In continuation with \cite{He1} and an unpublished paper by F.~Daumer \cite{Da}, we would like to explore the variation of the splitting for a Schr\"odinger operator coming from field theory when the potential changes as a function of the temperature from a one-well situation to a double well situation. We are for the moment only able to treat the fixed dimension case. But, because the most interesting thing would be to analyze the dependence on the dimension (in the perspective of applications to statistical mechanics) we have tried to follow some of the steps in the proof with respect to the dimension. All this analysis was strongly motivated by a course by M.~Kac where some heuristical discussion based on some Born-Oppenheimer analysis is presented. So our work can be considered as a first attempt for justifying this discussion.

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