  We study properties of the nodal sets of high frequency eigenfunctions and
quasimodes for radial perturbations of the Harmonic Oscillator. In particular,
we consider nodal sets on spheres of large radius (in the classically forbidden
region) for quasimodes with energies lying in intervals around a fixed energy
$E$. For well chosen intervals we show that these nodal sets exhibit
quantitatively different behavior compared to those of the unperturbed Harmonic
Oscillator. These energy intervals are defined via a careful analysis of the
eigenvalue spacings for the perturbed operator, based on analytic perturbation
theory and linearization formulas for Laguerre polynomials.
