Perturbative Semiclassical Trace Formulae for Harmonic Oscillators

Already 100 years ago Bohr used classical mechanics to build a model for a quantum system, the hydrogen atom. Although Bohr only included the circular classical orbits, the calculated energy spectrum from his model showed remarkable agreement with measurements. In this article we ex-pand the known semiclassical models by including more general perturba-tive 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 U(D) to O(D) symmetry breaking. We derive the gross structure of the semiclas-sical spectrum from periodic orbit theory, in the form of a perturbative (~ → 0) trace formula. We then show how to apply the results to even or-der 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.