Quantum Mushroom Billiards

Alex H. Barnett and Timo Betcke

work supported by the National Science Foundation

Preprints:

Quantum Mushroom Billiards, revised and expanded version, to appear, CHAOS, Dec 2007.

Here are movies of the 336 odd-symmetry eigenmodes in wavenumber range [90,100], computed using the scaling method. The mode number is around 1700. Only the right-half of the mushroom is shown:

Raw eigenmodes, with mode number j increasing in time. The intensity |φ_j|² is shown, with white zero and larger values darker. `Migrating scars' are actually visible in this movie, but only when you know what to look for. They are mush clearer in the following.
(5.4MB mpeg)

Local density of states over the same wavenumber interval. This is essentially the previous movie smoothed in wavenumber by a Gaussian of width 0.12. Look for the migrating scars moving down the middle of the left-hand wall (the symmetry axis). You may want to set your movie player on loop.
(2.8MB mpeg)

Migrating scars are believed to be a general phenomenon observed in the self-focal planes of reflective arcs (ie near the center of curvature), moving towards the self-focal point as wavenumber increases.

The classical (ray) paths contributing to this scar phenomenon are illustrated in the time domain by the following movie (10MB):

White has been used to highlight large ray phase space densities. Notice that nearly perfect refocusing of rays occurs at the launch point just after t=3.0 and t=6.0, but not at multiples of 3 after that.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-0507614

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