Band invariants for perturbations of the harmonic oscillator

We study the direct and inverse spectral problems for semiclassical operators of the form S = S0 + ¯h2V , where S0 = 12 (−¯h2 Rn + |x|2) is the harmonic oscillator and V :Rn→R 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). Published by Elsevier Inc.