  In this article we extend previous semiclassical studies by including more
general perturbative potentials of the harmonic oscillator in arbitrary spatial
dimensions. Our starting point is a radial harmonic potential with an arbitrary
even monomial perturbation, which we use to study the resulting $\mathrm{U}(D)$
to $\mathrm{O}(D)$ symmetry breaking. We derive the gross structure of the
semiclassical spectrum from periodic orbit theory, in the form of a
perturbative ($\hbar \rightarrow 0$) trace formula. We then show how to apply
the results to even order polynomial potentials, possibly including mean-field
terms. We have drawn the conclusion that the gross structure of the quantum
spectrum is determined from only classical circular- and diameter-orbits for
this class of systems.
