Band invariants for perturbations of the harmonic oscillator

We study the direct and inverse spectral problems for semiclassical operators of the form S=S0+ℏ2V, where is the harmonic oscillator and V:Rn→R is a tempered smooth function. We show that the spectrum of S forms eigenvalue clusters as ℏ tends to zero, and compute the first two associated “band invariants”. We derive several inverse spectral results for V, under various assumptions. In particular we prove that, in two dimensions, generic analytic potentials that are even with respect to each variable are spectrally determined (up to a rotation).