$L^p$ norms, nodal sets, and quantum ergodicity
Analysis of PDEs
2015-03-31 v2 Mathematical Physics
Dynamical Systems
math.MP
Spectral Theory
Abstract
For small range of , we improve the 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.
Cite
@article{arxiv.1411.4078,
title = {$L^p$ norms, nodal sets, and quantum ergodicity},
author = {Hamid Hezari and Gabriel Riviere},
journal= {arXiv preprint arXiv:1411.4078},
year = {2015}
}
Comments
27 pages. Appendix B on toral eigenfunctions is removed from the original posting. The background on L^p norms and nodal sets is updated