The eigenfunctions ei〈λ,x〉 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 ||φk||Lp ≥ C(∈)λkp-2/4p-∈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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics