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

Brack, M; Ögren, Magnus LU ; Yu, Yongle LU and Reimann, Stephanie LU (2005) In Journal of Physics A: Mathematical and General 38(46). p.9941-9967
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 1/4 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 e in semiclassical perturbation theory by integration of the action of the perturbed periodic HO orbit families over the manifold CP2 which is covered by the parameters describing their four-fold degeneracy. Then, we obtain an analytical uniform trace formula for arbitrary E 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)... (More)
We develop a uniform semiclassical trace formula for the density of states of a three-dimensional isotropic harmonic oscillator (HO), perturbed by a term 1/4 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 e in semiclassical perturbation theory by integration of the action of the perturbed periodic HO orbit families over the manifold CP2 which is covered by the parameters describing their four-fold degeneracy. Then, we obtain an analytical uniform trace formula for arbitrary E 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 E (or energy) -> 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 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) proportional to r(4). (Less)
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organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal of Physics A: Mathematical and General
volume
38
issue
46
pages
9941 - 9967
publisher
IOP Publishing
external identifiers
  • wos:000233696200007
  • scopus:27844470155
ISSN
0305-4470
DOI
10.1088/0305-4470/38/46/004
language
English
LU publication?
yes
id
ab72fd2b-0f9c-484b-9f53-607b7684599d (old id 211535)
date added to LUP
2007-09-21 09:16:10
date last changed
2017-02-26 04:05:55
@article{ab72fd2b-0f9c-484b-9f53-607b7684599d,
  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 1/4 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 e in semiclassical perturbation theory by integration of the action of the perturbed periodic HO orbit families over the manifold CP2 which is covered by the parameters describing their four-fold degeneracy. Then, we obtain an analytical uniform trace formula for arbitrary E 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 E (or energy) -> 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 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) proportional to r(4).},
  author       = {Brack, M and Ögren, Magnus and Yu, Yongle and Reimann, Stephanie},
  issn         = {0305-4470},
  language     = {eng},
  number       = {46},
  pages        = {9941--9967},
  publisher    = {IOP Publishing},
  series       = {Journal of Physics A: Mathematical and General},
  title        = {Uniform semiclassical trace formula for U(3) -> SO(3) symmetry breaking},
  url          = {http://dx.doi.org/10.1088/0305-4470/38/46/004},
  volume       = {38},
  year         = {2005},
}