$L^p$ 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.

arXiv comment27 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 (7)

• Lower bounds for nodal sets of eigenfunctions
• The quantum unique ergodicity conjecture for thin sets
• Small scale quantum ergodicity in negatively curved manifolds
• Local and global analysis of eigenfunctions
• The semiclassical zeta function for geodesic flows on negatively curved manifolds
• 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 (9)

• Small scale quantum ergodicity in negatively curved manifolds
• Planck-scale mass equidistribution of toral Laplace eigenfunctions
• Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces
• Improved critical eigenfunction restriction estimates on Riemannian manifolds with constant negative curvature
• Growth of high $L^p$ norms for eigenfunctions: an application of geodesic beams
• Applications of small scale quantum ergodicity in nodal sets
• Localized $L^p$-estimates of eigenfunctions: A note on an article of Hezari and Rivière
• Log-scale equidistribution of zeros of quantum ergodic eigensections
• Average $L^q$ growth and nodal sets of eigenfunctions of the Laplacian on surfaces