A semi-classical inverse problem I: Taylor expansions

Abstract

In dimension 1, we show that the Taylor expansion of a “generic” potential near a nondegenerate 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 does. 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.

Mathematics Subject Classification (2010): 34E20, 81Q10, 81Q20