Band Invariants for Perturbations of the Harmonic Oscillator

We study the direct and inverse spectral problems for semiclassical operators of the form S = S0 + ~V , where S0 = 12 ( −~∆Rn + |x| ) is the harmonic oscillator and V : R → R is a tempered smooth function. We show that the spectrum of S forms eigenvalue clusters as ~ 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).