Lp norms of eigenfunctions in the completely integrable case

John A. Toth*, Steve Zelditch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)343-368
Number of pages26
JournalAnnales Henri Poincare
Volume4
Issue number2
DOIs
StatePublished - 2003

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics

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