Weyl’s conjecture gives us the asymptotic distribution of the eigenvalues $\lambda$ of the Laplace operator in a domain $\Omega$ with homogeneous Dirichlet boundary condition on $\partial \Omega$.

According to this conjecture, the asymptotic distribution of the eigenvalues depends only on the volume and perimeter of the domain. However, I recall having read somewhere that “chaotic” domains (like billiards) support more eigenvalues in the same interval $[\lambda_1,\lambda_2]$ than “regular” domains (like squares, rectangles, circles etc.).

Is this qualitative, somehow vague statement true in any sense? If so, is this due to the degeneracy of some eigenvalues in regular domains, or are there other, “deeper” reasons? I have read that the statistics of the eigenvalue distribution depends on the form and the degree of “chaoticity” of the domain, so maybe this is a plausible reason. Any hint to the literature would be greatly appreciated!

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About scarring in different systems: the best understood case seems to be (quantum) cat maps, where you have both examples of scars (due to De Bievre, Faure and Nonnenmacher), and “non-scarring” results of Kurlberg-Rudnick. In the continuous case, the best chance for scarring is either Bunimovich stadium or Donnelly’s examples; in other situations (e.g. for Sinai billiard or in strictly negative curvature) you have much less chance. ]]>

A good question! I’m not entirely sure as to the answer. The thing is that (a) the scarred states will be very infrequent, as we know that almost all of the eigenstates do not exhibit scarring, and (b) there are also a large number of scarred quasimodes which behave physically like scarred states under either the wave equation or Schrodinger equation for bounded amounts of time, and could thus “spoof” the scarred states on these intervals. However it may be that the very long time singularity propagation behaviour of either the wave or Schrodinger equation may hinge on the existence of scarred states; if there are many such states one could imagine that a wave which is initially singular on a periodic trajectory (e.g. take a horizontal wave front moving vertically) will retain some residual version of that singularity for indefinite periods of time, whereas in the absence of scarring this singularity will eventually be “evenly distributed” across phase space, whatever that means. I don’t have a rigorous version of this statement though.

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