Semi-Classical Analysis for the Transfer Operator: Wkb Constructions in Dimension 1

Abstract

This lecture is devoted to problems coming from statistical mechanics. The transfer matrix approach consists in reducing the analysis of asymptotic properties of the energy of some statistical systems expressed in terms of Laplace integrals to the analysis of the spectral properties of transfer operators (matrices) appearing in the form of integral operators. Sometimes the new problem appears to have a semi-classical nature. Although the problem is similar to the semi-classical study of the Kac’s operator presented in our paper with M.Brunaud [BruHe91] which was devoted to the study of expexp(− v/2) · exp(h2∆) · exp(− v/2) for h small, new features appear for the model exp(− v/2h) · exp(h∆) · exp(− v/2h). The principal results we want to present concern the semi-classical analysis for this second operator. This lecture will only explain the constructions in the one-dimensional case. The proof of analogous results in dimension > 1 and further developments in large dimension will be given in [He95c]. (See also [He95a] and [He95b].)