Analytic Microlocal Bohr-Sommerfeld Expansions

Abstract: This article is devoted to Gevrey-analytic estimates in , of the Bohr-Sommerfeld expansion of the eigenvalues of self-adjoint pseudo-differential operators acting on L^2(R) in the regular case. We consider an interval of energies in which the spectrum of P is discrete and such that the energy sets are regular connected curves. Under some assumptions on the holomorphy of the symbol p, we will use the isometry between L^2(R) and the Bargmann space to obtain an exponentially sharp description of the spectrum in the energy window . More precisely it is possible to build exponentially sharp WKB quasimodes in the Bargmann space. A precise examination of the principal symbols will provide an interpretation to the Maslov correction $\pi$ in the Bargmann space.