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.