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Last year, as part of my “open problem of the week” series (now long since on hiatus), I featured one of my favorite problems, namely that of establishing scarring for the Bunimovich stadium.  I’m now happy to say that this problem has been solved (for generic stadiums, at least, and for phase space scarring rather than physical space scarring) by my old friend (and fellow Aussie), Andrew Hassell, in a recent preprint.  Congrats Andrew!

Actually, the argument is beautifully simple and short (the paper is a mere 9 pages), though it of course uses the basic theory of eigenfunctions on domains, such as Weyl’s law, and I can give the gist of it here (suppressing all technical details).

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This is a well known problem (see for instance this survey) in the area of “quantum chaos” or “quantum unique ergodicity”; I am attracted to it both for its simplicity of statement (which I will get to eventually), and also because it focuses on one of the key weaknesses in our current understanding of the Laplacian, namely is that it is difficult with the tools we know to distinguish between eigenfunctions (exact solutions to -\Delta u_k = \lambda_k u_k) and quasimodes (approximate solutions to the same equation), unless one is willing to work with generic energy levels rather than specific energy levels.

The Bunimovich stadium \Omega is the name given to any planar domain consisting of a rectangle bounded at both ends by semicircles. Thus the stadium has two flat edges (which are traditionally drawn horizontally) and two round edges, as this picture from Wikipedia shows:

Bunimovich stadium - Wikipedia

Despite the simple nature of this domain, the stadium enjoys some interesting classical and quantum dynamics. The classical dynamics, or billiard dynamics on \Omega is ergodic (as shown by Bunimovich) but not uniquely ergodic. In more detail: we say the dynamics is ergodic because a billiard ball with randomly chosen initial position and velocity (as depicted above) will, over time, be uniformly distributed across the billiard (as well as in the energy surface of the phase space of the billiard). On the other hand, we say that the dynamics is not uniquely ergodic because there do exist some exceptional choices of initial position and velocity for which one does not have uniform distribution, namely the vertical trajectories in which the billiard reflects orthogonally off of the two flat edges indefinitely.

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