Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2005-05-30
J. Phys. A: Math. Gen. 38 (2005) 9941-9967
Nonlinear Sciences
Exactly Solvable and Integrable Systems
LaTeX (revtex4), 26pp., 5 figures, 1 table; final version to be published in J. Phys. A (without appendices C and D)
Scientific paper
10.1088/0305-4470/38/46/004
We develop a uniform semiclassical trace formula for the density of states of a three-dimensional isotropic harmonic oscillator (HO), perturbed by a term $\propto r^4$. 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 in semiclassical perturbation theory by integration of the action of the perturbed periodic HO orbits over the manifold $\mathbb{C}$P$^2$ which characterizes their 4-fold degeneracy. Then we obtain an analytical uniform trace formula 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 $\omega_r:\omega_\phi=N:M$ of radial and angular frequencies. The same holds also for the limit of a purely quartic spherical potential $V(r)\propto r^4$.
Brack Matthias
Ogren Magnus
Reimann Stephanie M.
Yu Yue
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