Analytic Microlocal Bohr-Sommerfeld Expansions

TL;DR

This paper develops Gevrey-analytic estimates for the Bohr-Sommerfeld expansion of eigenvalues of self-adjoint pseudo-differential operators, providing exponentially sharp spectral descriptions using Bargmann space techniques.

Contribution

It introduces a method to obtain exponentially sharp Bohr-Sommerfeld expansions for eigenvalues via Gevrey-analytic estimates and Bargmann space analysis.

Findings

Exponential accuracy in spectral estimates achieved.

Construction of exponentially sharp WKB quasimodes.

Interpretation of Maslov correction in Bargmann space.

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.