$L^p$ norms, nodal sets, and quantum ergodicity

Abstract

For small range of $p>2$, we improve the $L^p$ bounds of eigenfunctions of the Laplacian on negatively curved manifolds. Our improvement is by a power of logarithm for a full density sequence of eigenfunctions. We also derive improvements on the size of the nodal sets. Our proof is based on a quantum ergodicity property of independent interest, which holds for families of symbols supported in balls whose radius shrinks at a logarithmic rate.

27 pages. Appendix B on toral eigenfunctions is removed from the original posting. The background on L^p norms and nodal sets is updated

References in corpus (3)

• Local and global analysis of eigenfunctions
• Dispersion and controllability for the Schrödinger equation on negatively curved manifolds
• Localized $L^p$-estimates of eigenfunctions: A note on an article of Hezari and Rivière

Cited by in corpus (1)

• Localized $L^p$-estimates of eigenfunctions: A note on an article of Hezari and Rivière