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Uniform semiclassical trace formula for U(3) --> SO(3) symmetry breaking

Brack, Matthias and Ögren, M. and Yu, Y. and Reimann, Stephanie (2005) Uniform semiclassical trace formula for U(3) --> SO(3) symmetry breaking. Journal of Physics A 38, pp. 9941-9967.

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Other URL: http://ej.iop.org/links/q22/5zfdYXxqoSK6BbmsLjOx4g/a5_46_004.pdf

Abstract

We develop a uniform semiclassical trace formula for the density of states of a three-dimensional isotropic harmonic oscillator (HO), perturbed by a term $\\frac14 \\epsilon\\,r^4$ . This term breaks the U(3) symmetry of the HO, resulting in a spherical system with SO(3) symmetry. We first treat the anharmonic term for small $\\epsilon$ in semiclassical perturbation theory by integration of the action of the perturbed periodic HO orbit families over the manifold $\\mathbb{C}$ P $^2$ which is covered by the parameters describing their four-fold degeneracy. Then we obtain an analytical uniform trace formula for arbitrary $\\epsilon$ which in the limit of strong perturbations (or high energy) asymptotically goes over into the correct trace formula of the full anharmonic system with SO(3) symmetry, and in the limit $\\epsilon$ (or energy) $\\to 0$ restores the HO trace formula with U(3) symmetry. We demonstrate that the gross-shell structure of this anharmonically perturbed system is dominated by the two-fold degenerate diameter and circular orbits, and ${\\it not}$ by the orbits with the largest classical degeneracy, which are the three-fold degenerate tori with rational ratios $\\omega_r:\\omega_\\varphi=N\\!:\\!M$ of radial and angular frequencies. The same holds also for the limit of a purely quartic spherical potential $V(r)\\propto r^4$ .