An Inverse Problem for Trapping Point Resonances

We consider semi-classical Schr{\"o}dinger operator $P(h)=-h^2\Delta +V(x)$ in ${\mathbb R}^n$ such that the analytic potential $V$ has a non-degenerate critical point $x_0=0$ with critical value $E_0$ and we can define resonances in some fixed neighborhood of $E_0$ when $h>0$ is small enough. If the eigenvalues of the Hessian are $\zz$-independent the resonances in $h^\delta$-neighborhood of $E_0$ ($\delta >0$) can be calculated explicitly as the eigenvalues of the semi-classical 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 $x_0.$ As a consequence, the resonances in a $h^\delta$-neighborhood of $E_0$ determine the first $N$ terms in the Taylor series of $V$ at $x_0.$ The proof uses the recent inverse spectral results of V. Guillemin and A. Uribe.