Band invariants for perturbations of the harmonic oscillator

We study the direct and inverse spectral problems for semiclassical operators of the form $S = S_0 +\h^2V$, where $S_0 = \frac 12 \Bigl(-\h^2\Delta_{\bbR^n} + |x|^2\Bigr)$ is the harmonic oscillator and $V:\bbR^n\to\bbR$ is a tempered smooth function. We show that the spectrum of $S$ forms eigenvalue clusters as $\h$ 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).