Bohr-Sommerfeld Rules to All Orders

Abstract

H on the real line. The formula gives the highorder terms in the expansion in powers of h of the semi-classical action using only integrals on the energy curves of quantities which are locally computable from the Weyl symbol. The recipe uses only the knowledge of the Moyal formula expressing the star product of Weyl symbols. It is important to note that our method assumes already the existence of Bohr-Sommerfeld rules to any order (which is usually shown using some precise Ansatz for the eigenfunctions, like the WKB-Maslov Ansatz) and the problem we address here is only about ways to compute these corrections. Existence of corrections to any order to Bohr-Sommerfeld rules is well known and can be found for example in [8] and [15] Section 4.5. Our way to get these high-order corrections is inspired by A. Voros’s thesis (1977) [13], [14]. The reference [1], where a very similar method is sketched, was given to us by A. Voros. We use also in an essential way the nice formula of