An Inverse Problem for Trapping Point Resonances

We consider semi-classical Schr¨odinger operator P(h)=−h2+V (x) in Rn such that the analytic potential V has a non-degenerate critical point x0 = 0 with critical value E0 and we can define resonances in some fixed neighborhood of E0 when h > 0 is small enough. If the eigenvalues of the Hessian are Z-independent the resonances in hδ-neighborhood of E0 (δ>0) can be calculated explicitly as the eigenvalues of the semiclassical Birkhoff normal form. Assuming that potential is symmetric with respect to reflections about the coordinate axes we show that the classical Birkhoff normal form determines the Taylor series of the potential at x0. As a consequence, the resonances in a hδ-neighborhood of E0 determine the first N terms in the Taylor series of V at x0. The proof uses the recent inverse spectral results of V. Guillemin and A. Uribe