Lp Norms of Eigenfunctions in the Completely Integrable Case
By J. Toth, Steve Zelditch. 2002
48 citations recorded by the source corpus.
Abstract. The eigenfunctions $ e^{i\langle\lambda,x\rangle} $ of the Laplacian on a flat torus have uniformly bounded Lp norms. In this article, we prove that for every other quantum integrable Laplacian, the Lp norms of the joint eigenfunctions blow up at least at the rate $ \| \varphi_k \| L^{p} \geq C(\epsilon)\lambda_{k}^{{p-2\over4p}-\epsilon} $ when p > 2. This gives a quantitative refinement of our recent result [TZ1] that some sequence of eigenfunctions must blow up in Lp unless (M,g) is flat. The better result in this paper is based on mass estimates of eigenfunctions near singular leaves of the Liouville foliation.
Subfields: Nuclear and High Energy Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/3340e6d832342a73a82cc518e8e66539713776f5
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In Nuclear and High Energy Physics
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Citations in the corpus
- On Quantum Unique Ergodicity for Locally Symmetric Spaces (2004, background): “DICITY FOR LOCALLY SYMMETRIC SPACES I 32 From the microlocal viewpoint the system under consideration resembles completely integrable systems, in that there are several commuting observable; see e.g. [20]. Of course, the Cartan flow differs from th e completely integrable case in that it is very chao”
- Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains (2006, unknown): “[42].”
- Kakeya-Nikodym averages and $L^p$-norms of eigenfunctions (2009, background): “!1 1=2 Z je j j2ds>0; then would have to be part of a stable smoothly closed geodesic, and not just a closed geodesic as we showed above. Toth and Zeldtich made a similar conjecture to this in [34], saying that, in n-dimensions, if is a closed stable geodesic then one should be able to nd a seq”