


Bunimovich stadium - Wikipedia


Rather than working with (classical) individual trajectories, one can also work with (classical) invariant ensembles – probability distributions in phase space which are invariant under the billiard dynamics. Ergodicity then says that (at a fixed energy) there are no invariant absolutely continuous ensemble other than the obvious one, namely the probability distribution with uniformly distributed position and velocity direction. On the other hand, unique ergodicity would say the same thing but dropping the “absolutely continuous” – but each vertical bouncing ball mode creates a singular invariant ensemble along that mode, so the stadium is not uniquely ergodic.

Now from physical considerations we expect the quantum dynamics of a system to have similar qualitative properties as the classical dynamics; this can be made precise in many cases by the mathematical theories of semi-classical analysis and microlocal analysis. The quantum analogue of the dynamics of classical ensembles is the dynamics of the Schrödinger equation i\hbar \partial_t \psi + \frac{\hbar^2}{2m} \Delta \psi = 0, where we impose Dirichlet boundary conditions (one can also impose Neumann conditions if desired, the problems seem roughly the same). The quantum analogue of an invariant ensemble is a single eigenfunction -\Delta u_k = \lambda_k u_k, which we normalise in the usual L^2 manner, so that \int_\Omega |u_k|^2 = 1. (Due to the compactness of the domain \Omega, the set of eigenvalues \lambda_k of the Laplacian -\Delta is discrete and goes to infinity, though there is some multiplicity arising from the symmetries of the stadium. These eigenvalues are the same eigenvalues that show up in the famous “can you hear the shape of a drum?” problem.) Roughly speaking, quantum ergodicity is then the statement that almost all eigenfunctions are uniformly distributed in physical space (as well as in the energy surface of phase space), whereas quantum unique ergodicity (QUE) is the statement that all eigenfunctions are uniformly distributed. In particular:

  • If quantum ergodicity holds, then for any open subset A \subset \Omega we have \int_A |u_k|^2 \to |A|/|\Omega| as \lambda_k \to \infty, provided we exclude a set of exceptional k of density zero.
  • If quantum unique ergodicity holds, then we have the same statement as before, except that we do not need to exclude the exceptional set.

(In fact, quantum ergodicity and quantum unique ergodicity say somewhat stronger things than the above two statements, but I would need tools such as pseudodifferential operators to describe these more technical statements, and so I will not do so here.)

Now it turns out that for the stadium, quantum ergodicity is known to be true; this specific result was first obtained by Gérard and Leichtman, although “classical ergodicity implies quantum ergodicity” results of this type go back to Schnirelman (see also Zelditch and Colin de Verdière). These results are established by microlocal analysis methods, which basically proceed by aggregating all the eigenfunctions together into a single object (e.g. a heat kernel, or some other function of the Laplacian) and then analysing the resulting aggregate semiclassically. It is because of this aggregation that one only gets to control almost all eigenfunctions, rather than all eigenfunctions. Here is a picture of a typical eigenfunction for the stadium (from Douglas Stone’s page):

Typical stadium eigenfunction

In analogy to the above theory, one generally expects classical unique ergodicity should correspond to QUE. For instance, there is the famous (and very difficult) quantum unique ergodicity conjecture of Rudnick and Sarnak, which asserts that QUE holds for all compact manifolds without boundary with negative sectional curvature. This conjecture will not be discussed here (it would warrant an entire post in itself, and I would not be the best placed to write it). Instead, we focus on the Bunimovich stadium. The stadium is clearly not classically uniquely ergodic due to the vertical bouncing ball modes, and so one would conjecture that it is not QUE either. In fact one conjectures the slightly stronger statement:

  • Scarring conjecture: there exists a subset A \subset \Omega and a sequence u_{k_j} of eigenfunctions with \lambda_{k_j} \to\infty, such that \int_A |u_{k_j}|^2 does not converge to |A|/|\Omega|. Informally, the eigenfunctions either concentrate (or “scar”) in A, or on the complement of A.

Indeed, one expects to take A to be a union of vertical bouncing ball trajectories (from Egorov’s theorem (in microlocal analysis, not the one in real analysis), this is almost the only choice). This type of failure of QUE even in the presence of quantum ergodicity has already been observed for some simpler systems, such as the Arnold cat map. Some further discussion of this conjecture can be found here. Here are some pictures from Arnd Bäcker‘s page of some eigenfunctions (displaying just one quarter of the stadium to save space) which seem to exhibit scarring:

Scarring eigenfunctions

Of course, each of these eigenfunctions has a fixed finite energy, and so these numerics do not directly establish the scarring conjecture, which is a statement about the asymptotic limit as the energy becomes infinite.

One reason this conjecture appeals to me (apart from all the gratuitous pretty pictures one can mention while discussing it) is that there is a very plausible physical argument, due to Heller and refined by Zelditch, which indicates the conjecture is almost certainly true. Roughly speaking, it runs as follows. Using the rectangular part of the stadium, it is easy to construct (high-energy) quasimodes of order 0 which scar (concentrate on a proper subset A of \Omega) – roughly speaking, these are solutions u to an approximate eigenfunction equation -\Delta u = (\lambda + O(1)) u for some \lambda. For instance, if the two horizontal edges of the stadium lie on the lines y=0 and y=1, then one can take u(x,y) = \varphi(x) \sin(\pi n y) and \lambda = \pi^2 n^2 for some large integer n and some suitable bump function \varphi. Using the spectral theorem, one expects u to concentrate its energy in the band {}[\pi^2 n^2 - O(1), \pi^2 n^2 + O(1)]. On the other hand, in two dimensions the Weyl law for distribution of eigenvalues asserts that the eigenvalues have an average spacing comparable to 1. If (and this is the non-rigorous part) this average spacing also holds on a typical band {}[\pi^2 n^2 - O(1), \pi^2 n^2 + O(1)], this shows that the above quasimode is essentially generated by only O(1) eigenfunctions. Thus, by the pigeonhole principle (or more precisely, Pythagoras’ theorem), at least one of the eigenfunctions must exhibit scarring.

[Update, Mar 28: As Greg Kuperberg pointed out, I oversimplified the above argument. The quasimode is so weak that the eigenfunctions that comprise it could in fact spread out (as per the uncertainty principle) and fill out the whole stadium. However, if one looks in momentum space rather than physical space, the scarring of the quasimode is so strong that it must persist to one of the eigenfunctions, leading to failure of QUE even if this may not quite be detectable purely in the physical space sense described above.]

The big gap in this argument is that nobody knows how to take the Weyl law (which is proven by the microlocal analysis approach, i.e. aggregate all the eigenstates together and study the combined object) and localise it to such an extremely sparse set of narrow energy bands. (Using the standard error term in Weyl’s law one can localise to bands of width O(n) around, say, \pi^2 n^2, and by using the ergodicity one can squeeze this down to o(n), but to even get control on a band of with width O(n^{1-\epsilon}) would require a heroic effort (analogous to establishing a zero-free region \{ s: \hbox{Re}(s) > 1-\epsilon\} for the Riemann zeta function). The enemy is somehow that around each energy level \pi^2 n^2, a lot of exotic eigenfunctions spontaneously appear, which manage to dissipate away the bouncing ball quasimodes into a sea of quantum chaos. This is exceedingly unlikely to happen, but we do not seem to have tools available to rule it out.

One indication that the problem is not going to be entirely trivial is that one can show (basically by unique continuation or control theory arguments) that no pure eigenfunction can be solely concentrated within the rectangular portion of the stadium (where all the vertical bouncing ball modes are); a significant portion of the energy must leak out into the two “wings” (or at least into arbitrarily small neighbourhoods of these wings). This was established by Burq and Zworski.

On the other hand, the stadium is a very simple object – it is one of the simplest and most symmetric domains for which we cannot actually compute eigenfunctions or eigenvalues explicitly. It is tempting to just discard all the microlocal analysis and just try to construct eigenfunctions by brute force. But this has proven to be surprisingly difficult; indeed, despite decades of sustained study into the eigenfunctions of Laplacians (given their many applications to PDE, to number theory, to geometry, etc.) we still do not know very much about the shape and size of any specific eigenfunction for a general manifold, although we know plenty about the average-case behaviour (via microlocal analysis) and also know the worst-case behaviour (by Sobolev embedding or restriction theorem type tools). This conjecture is one of the simplest conjectures which would force us to develop a new tool for understanding eigenfunctions, which could then conceivably have a major impact on many areas of analysis.

One might consider modifying the stadium in order to make scarring easier to show, for instance by selecting the dimensions of the stadium appropriately (e.g. obeying a Diophantine condition), or adding a potential or magnetic term to the equation, or perhaps even changing the metric or topology. To have even a single rigorous example of a reasonable geometric operator for which scarring occurs despite the presence of quantum ergodicity would be quite remarkable, as any such result would have to involve a method that can deal with a very rare set of special eigenfunctions in a manner quite different from the generic eigenfunction.

Actually, it is already interesting to see if one can find better quasimodes than the ones listed above which exhibit scarring, i.e. to improve the O(1) error in the spectral bandwidth. My good friend Maciej Zworski has offered a dinner in a good French restaurant for this precise problem, as well as a dinner in a very good French restaurant for the full scarring conjecture. (While I may not know as many three-star restaurants as Maciej, I can certainly offer a nice all-expenses-paid trip to sunny Los Angeles for anyone who achieves a breakthrough on any of the open problems listed here. ;-) ).