A semi - classical inverse problem II : reconstruction of the potential

This paper is the continuation of [4], where Victor Guillemin and I proved the following result: the Taylor expansion of the potential V (x) (x ∈ R) at a non degenerate critical point x0 of V , satisfying V (x0) 6= 0, is determined by the semi-classical spectrum of the associated Schrödinger operator near the corresponding critical value V (x0). Here, I prove results which are stronger in some aspects: the potential itself, without any analyticity assumption, but with some genericity conditions, is determined from the semi-classical spectrum. Moreover, our method gives an explicit way to reconstruct the potential. Inverse spectral results for Sturm-Liouville operators are due to Borg, Gelfand, Levitan, Marchenko and others (see for example [8]). They need the spectra of the differential operator with two different boundary conditions in order to recover the potential. Our results are different in several aspects: