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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
PubMed
Summary

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.

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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.

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  • 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.