In France we have a very useful saying “le ridicule ne tue pas” (being ridiculous does not kill), so I’ll attempt to clear my mind on a few topics here, and come what may… 🙂
The first thing that puzzles me is an answer of Terry Tao on his blog to a silly question of mine about eigenfunctions of the Dirichlet Laplacian on a general manifold . He said that norms describe the distributions of the eigenfunctions and that their understanding is still far from satisfactory.
Let’s pick an eigenfunction . I didn’t see at all how a countable sequence of positive numbers, namely the , were going to tell us anything about for any . Could one reconstruct from ? (I heard in algebraic geometry people define objects as the union over all rings of zeros of a given polynomial, maybe there’s a similarity here, we look at in all the . But I digress…)
So I started digging the arXiv and found a few papers.
A first one is by Zelditch and Toth here which deals with the compact completely integrable case. It looks quite readable, I’ll try in a few months. Their main result (which is sharp in its setting) is:
Theorem (Toth-Zelditch): Let be a Riemannian manifold with completely integrable geodesic flow satisfying Eliasson’s non-degeneracy condition. Then, unless is a flat torus, for every there exists a sequence of eigenfunctions satisfying and (for …).
So this partially answers my question: the norms of the eigenfunctions blow up as a power of the corresponding eigenvalue. But again, what does this tell us on the distribution of the eigenfunctions, other that saying they somehow blow-up ? It could happen in so many ways, no ?
Another paper is by Schiffman, Tate and Zelditch and deals with some completely integrable systems in complex geometry. The results they obtain there are finer, since do speak of the eigenfunctions themselves by obtaining pointwise estimates. I’m not knowledgeable in complex geometry, but the result that the eigenfunctions behave like gaussians centered around a classical torus was certainly known by physists for a long time, so this is all very sensible and interesting.
The second thing I’d like to understand is related to separation of variables, more precisely to Tao’s question in one of his comments:
- one can phrase this problem as a question of determining how stable the method of separation of variables is in solving PDE. This method trivially lets us determine the eigenfunctions of a rectangle; why shouldn’t some variant of it still give some control on eigenfunctions of a near-rectangle?
This is what leads me to conformal mappings. Riemann’s theorem implies that there exists a conformal mapping of the rectangle to the unit disk, and also a conformal mapping of any near-rectangle to the unit disk. I’m not a friend of complex analysis, but provided can be inverted, the map maps the rectangle’s nice separating coordinate system to that of the near-rectangle.
So the question is what does the Laplacian become under this transformation, i.e. are there nasty terms coming in ? Comments welcomed.