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Updated: Aug 9, 2025

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Scarring in Rough Rectangular Billiards.

Felix M Izrailev1,2, German A Luna-Acosta1, J A Mendez-Bermudez1

  • 1Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico.

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|February 25, 2023
PubMed
Summary
This summary is machine-generated.

Eigenstate scarring in corrugated billiards differs from known models, revealing two distinct scar state sets. Surface roughness influences wave scattering, with symmetry playing a key role.

Keywords:
localization effectsquantum billiardsrough billiardsscars

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Area of Science:

  • Quantum mechanics
  • Wave phenomena
  • Mathematical physics

Background:

  • Billiards with corrugated surfaces exhibit unique quantum mechanical properties.
  • Scarring of eigenstates in quantum billiards is a well-studied phenomenon, particularly in Sinai and Bunimovich billiards.
  • Understanding wave scattering in quasi-one-dimensional waveguides is crucial for device applications.

Purpose of the Study:

  • To investigate the mechanism of eigenstate scarring in rectangular billiards with corrugated surfaces.
  • To identify and characterize different sets of scar states.
  • To analyze the influence of surface roughness symmetry on eigenstate behavior and wave scattering.

Main Methods:

  • Numerical analysis of eigenstates in corrugated billiards.
  • Comparison with classical bouncing ball trajectories and plane-wave states.
  • Reduction of a one-particle model to an effective two-particle system with interaction.
  • Investigation of eigenstate repulsion from rough surfaces.

Main Results:

  • Two distinct sets of scar states were identified: one linked to classical trajectories, another emerging in momentum space.
  • Eigenstates exhibit repulsion from corrugated surfaces, with effects modulated by the symmetry of roughness.
  • The repulsion effect is significant, impacting all eigenstates and highlighting the importance of surface profile symmetry.
  • A novel theoretical approach was developed by reducing the problem to an effective two-particle system.

Conclusions:

  • The scarring mechanism in corrugated billiards is distinct from other billiard models.
  • Surface roughness symmetry is critical for controlling wave scattering phenomena in these systems.
  • The findings have implications for understanding electromagnetic and electron wave scattering in waveguides.
  • The developed two-particle model provides a new framework for analyzing corrugated billiards.