Norms of modes and quasi-modes revisited

Abstract

This article is devoted to the analysis of eigenfunctions (modes) and approx- imate eigenfunctions (quasi-modes) of the Laplacian on a compact manifold (M, g) with completely integrable geodesic flow. We give a new proof of the main result of (TZ) that (M, g) with integrable Laplacians and with uniformly bounded eigenfunctions must be flat. The proof is based on the use of Birkhoff normal forms and on a comparison of modes and quasimodes. In the process, we discuss tunnelling between resonant tori and give a proof that eigenfunctions concentrate on individual tori in the non-resonant case. We also give brief expositions of results in (TZ2, SZ).

Locations

• Contemporary mathematics - American Mathematical Society - View
• arXiv (Cornell University) - View - PDF