Uniform semiclassical trace formula for U(3) -> SO(3) symmetry breaking
(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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/211535
- author
- Brack, M ; Ögren, Magnus LU ; Yu, Yongle LU and Reimann, Stephanie LU
- organization
- publishing date
- 2005
- 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
- additional info
- The information about affiliations in this record was updated in December 2015. The record was previously connected to the following departments: Mathematical Physics (Faculty of Technology) (011040002)
- id
- ab72fd2b-0f9c-484b-9f53-607b7684599d (old id 211535)
- date added to LUP
- 2016-04-01 15:22:15
- date last changed
- 2022-03-30 00:58:57
@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}},
doi = {{10.1088/0305-4470/38/46/004}},
volume = {{38}},
year = {{2005}},
}