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Semiclassical trace formula for the two-dimensional radial power-law potentials.
A G Magner1, A A Vlasenko, K Arita
1Institute for Nuclear Research, 03680 Kiev, Ukraine. magner@kinr.kiev.ua
Summary
A new trace formula accurately describes quantum spectra in 2D power-law potentials, even near symmetry breaking points. This method enhances understanding of shell structure in many-fermion systems.
Area of Science:
- Quantum mechanics
- Statistical physics
- Nuclear physics
Background:
- The Woods-Saxon potential models nuclear structure, but its quantum spectra are complex.
- Semiclassical methods offer approximations but often fail at critical points like bifurcations.
Purpose of the Study:
- To derive a trace formula for single-particle level density in 2D radial power-law potentials.
- To analyze the contribution of periodic orbits to level density near bifurcations.
- To validate the formula against quantum spectra and explore its limits.
Main Methods:
- Improved stationary phase method for trace formula derivation.
- Analytical calculations for power-law potentials with α=4 and α=6.
- Comparison of semiclassical results with quantum spectra for many-fermion systems.
Main Results:
- The derived trace formula accurately reproduces quantum spectra, including shell corrections.
- Periodic orbit contributions significantly impact level density near bifurcations.
- The formula remains valid through symmetry-breaking catastrophe points where standard methods fail.
Conclusions:
- The improved stationary phase method provides a robust tool for analyzing quantum systems.
- The trace formula offers insights into the fine shell structure of many-fermion systems.
- The formula unifies various physical limits, including harmonic oscillators and spherical billiards.
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