Bifurcations of periodic orbits and uniform approximations
- 7 July 1997
- journal article
- Published by IOP Publishing in Journal of Physics A: General Physics
- Vol. 30 (13) , 4537-4562
- https://doi.org/10.1088/0305-4470/30/13/010
Abstract
We derive uniform approximations for contributions to Gutzwiller's periodic-orbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space. There, orbits lie close together and give collective contributions, while the individual contributions of Gutzwiller's type would diverge at the bifurcation. New results for the tangent, the period-doubling and the period-tripling bifurcation are given. They are obtained by going beyond the local approximation and including higher-order terms in the normal form of the action. The uniform approximations obtained are tested on the kicked top and are found to be in excellent agreement with exact quantum results.Keywords
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This publication has 23 references indexed in Scilit:
- Semiclassical trace formulae for systems with non-Abelian symmetryJournal of Physics A: General Physics, 1992
- Semiclassical trace formulas in the presence of continuous symmetriesPhysical Review A, 1991
- Resonant periodic orbits and the semiclassical energy spectrumJournal of Physics A: General Physics, 1987
- Calculating the bound spectrum by path summation in action-angle variablesJournal of Physics A: General Physics, 1977
- Closed orbits and the regular bound spectrumProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 1976
- Solution of the Schrödinger equation in terms of classical pathsAnnals of Physics, 1974
- Distribution of eigenfrequencies for the wave equation in a finite domain: III. Eigenfrequency density oscillationsAnnals of Physics, 1972
- Periodic Orbits and Classical Quantization ConditionsJournal of Mathematical Physics, 1971
- INSTABILITY IN A HAMILTONIAN SYSTEM AND THE DISTRIBUTION OF ASTEROIDSMathematics of the USSR-Sbornik, 1970
- Generic bifurcation of periodic pointsTransactions of the American Mathematical Society, 1970