Some inverse spectral results for semi-classical Schr\"odinger operators
  We consider a semi-classical Schr\"odinger operator, -h^2\Delta + V(x). Assuming that the potential admits a unique global minimum and that the eigenvalues of the Hessian are linearly independent over the rationals, we show that the low-lying eigenvalues of the operator determine the Taylor series of the potential at the minimum. 
V. Guillemin and A. Uribe