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Localization of eigenfunctions in the stadium billiard.

W E Bies¹, L Kaplan, M R Haggerty

¹Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA. bies@blizzard.harvard.edu

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

|June 21, 2001

Summary

Symmetry effects significantly influence quantum localization in stadium billiards, exceeding predictions from random-matrix theory. This study reveals excess localization on periodic orbits and symmetry lines, driven by fundamental symmetries.

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Area of Science:

Quantum chaos
Statistical mechanics

Background:

The stadium billiard model is a key system for studying quantum chaos.
Understanding eigenstate localization is crucial for predicting system behavior.

Purpose of the Study:

To systematically investigate scarring and symmetry effects on eigenstate localization in the stadium billiard.
To compare localization phenomena with predictions from random-matrix theory.

Main Methods:

Analysis of individual eigenfunction localization in Husimi phase space.
Investigation of local density of states as a function of phase space location.
Quantitative analysis of symmetry influences (parity, time-reversal) on localization.

Main Results:

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Average localization exceeds random-matrix theory predictions, even after excluding bouncing-ball states and visible scars.

Excess localization is observed on short periodic orbits and symmetry-related lines.

Scarring is a consistent feature across all energy ranges studied.

Conclusions:

Symmetry considerations, including parity and time-reversal, are primary drivers of excess eigenstate localization.
The observed localization patterns are quantitatively explained and compared with analytical predictions.