A semi - classical inverse problem I : Taylor expansions . ( to the memory of Hans Duistermaat )

In dimension 1, we show that the Taylor expansion of a “generic” potential near a non degenerate critical point can be recovered from the knowledge of the semi-classical spectrum of the associated Schrödinger operator near the corresponding critical value. Contrary to the work of previous authors, we do not assume that the potential is even. The classical Birkhoff normal form does not contain enough information to determine the potential, but the quantum Birkhoff normal form does1. In a companion paper [5], the first author shows how the potential itself is, without any analyticity assumption and under some mild genericity hypotheses, determined by the semi-classical spectrum.