Some inverse spectral results for semi-classical Schrödinger operators

Abstract

We show that the Birkhoff normal form of a classical Hamiltonian $H(x,\xi) = \norm{\xi}^2+V(x)$ at a non-degenerate minimum $x_0$ of the potential determines the Taylor series of the potential at $x_0$, provided the eigenvalues of the Hessian are linearly independent over $\bbQ$ and $V$ satisfies a symmetry condition near $x_0$. As a consequence, if $x_0$ is the unique global minimum of $V$, the low-lying eigenvalues of the semi-classical Schr\odinger operator, $-\h^2\Delta + V(x)$, determine the Taylor series of the potential at $x_0$.