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Semiclassical catastrophe theory of simple bifurcations
1Institute for Nuclear Research, 03680 Kyiv, Ukraine.
Physical Review. E
|January 20, 2018
Summary
The improved stationary phase method (ISPM) enhances catastrophe theory for bifurcation problems. This method accurately predicts quantum results for radial power-law potentials near bifurcation points.
Area of Science:
- Mathematical Physics
- Theoretical Physics
- Quantum Mechanics
Background:
- Catastrophe theory and the stationary phase method are crucial for analyzing caustics and turning points.
- Bifurcation problems in Hamiltonian systems present significant analytical challenges.
Purpose of the Study:
- To extend Fedoriuk-Maslov catastrophe theory for solving bifurcation problems using the improved stationary phase method (ISPM).
- To develop trace formulas for radial power-law (RPL) potentials via ISPM.
- To analyze the contribution of parent and newborn orbits during bifurcation.
Main Methods:
- Application of the improved stationary phase method (ISPM) based on second- and third-order expansions of classical action.
- Development of trace formulas for radial power-law potentials.
- Analysis of Hamiltonian systems with continuous symmetries and bifurcation scenarios.
Main Results:
- Demonstrated significant enhancement of contributions from parent and newborn orbits at bifurcation points.
- Proposed an ISPM trace formula where parent orbit contributions are incorporated into newborn orbit terms.
- Achieved good agreement between ISPM-calculated level densities and quantum results for RPL potentials.
Conclusions:
- The ISPM effectively addresses bifurcation problems in Hamiltonian systems.
- The developed ISPM trace formula accurately captures the behavior of radial power-law potentials near and far from bifurcation points.
- ISPM provides a robust framework for analyzing complex quantum phenomena in physical systems.
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