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Diffractive corrections in the trace formula for polygonal billiards

Bogomolny¹, Pavloff, Schmit

¹Laboratoire de Physique Theorique et Modeles Statistiques,* Universite Paris-Sud, Bainsertion marktiment 100, F-91405 Orsay Cedex, France.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

|November 23, 2000

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This study introduces a new method to calculate spectral density in polygonal billiards by accounting for diffractive orbits. The approach significantly reduces semiclassical errors, improving accuracy for quantum chaos research.

Area of Science:

Physics
Quantum Chaos
Mathematical Physics

Background:

Spectral density calculations in billiards are crucial for understanding quantum chaos.
Diffractive orbits in polygonal billiards present unique challenges for semiclassical approximations.
Existing methods like Keller's geometrical theory of diffraction are insufficient for these complex scenarios.

Purpose of the Study:

To derive contributions to the trace formula for spectral density in two-dimensional polygonal billiards.
To accurately account for the role of diffractive orbits in these systems.
To develop an improved approximation beyond Keller's theory for diffraction corrections.

Main Methods:

Derivation of trace formula contributions incorporating diffractive orbits.

Related Experiment Videos

Development of an alternative approximation based on Kirchhoff's theory of diffraction.

Numerical checks to validate the accuracy of the proposed method.

Main Results:

The first diffractive correction to the periodic orbit expansion is of the same order as an isolated orbit.
A significant reduction (two orders of magnitude) in typical semiclassical error was achieved.
The method accurately treats flux-line diffraction, a related problem.

Conclusions:

The developed method provides a more accurate way to calculate spectral density in polygonal billiards.
Kirchhoff's theory offers a viable alternative to Keller's theory for specific diffraction corrections.
This work advances the understanding of quantum chaos in systems with complex boundaries.