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Anomalous shell effect in the transition from a circular to a triangular billiard.

Ken-Ichiro Arita¹, Matthias Brack

¹Department of Physics, Nagoya Institute of Technology, 466-8555 Nagoya, Japan.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|July 23, 2008

Summary

We studied chaotic dynamics in a billiard system transitioning from circular to triangular shapes. A significant shell effect was observed, linked to a specific periodic orbit bifurcation, and accurately modeled using semiclassical methods.

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Area of Science:

Physics
Quantum Mechanics
Dynamical Systems

Background:

Billiard systems are fundamental models for studying classical and quantum chaos.
Understanding shape transitions in dynamical systems is crucial for various physics fields.

Purpose of the Study:

To investigate the emergence of shell effects in a two-dimensional nonintegrable billiard system undergoing shape transition.
To analyze the role of periodic orbits and bifurcations in semiclassical dynamics.

Main Methods:

Application of periodic orbit theory and semiclassical analysis.
Utilizing Gutzwiller's semiclassical trace formula with global uniform approximation.
Studying a system smoothly deformed from circular to equilateral triangular boundaries.

Main Results:

Observed a pronounced shell effect during the shape transition, despite increasing classical chaos.
Identified a codimension-2 bifurcation of the triangular periodic orbit as the source of the shell effect.
Demonstrated that the semiclassical trace formula accurately describes the quantum mechanical level density.

Conclusions:

The shell effect in this system is a robust phenomenon driven by specific orbital dynamics.
Semiclassical theory provides an accurate framework for understanding quantum properties of chaotic systems.
Discrete symmetries play a role in enhancing the observed shell effect.