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

Hamid Hezari, Gabriel Riviere

Published 2014-11-14Version 1

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. In Appendix B, we show that, in the case of a rational torus, this quantum ergodicity property still holds for symbols supported in balls with a radius shrinking at a polynomial rate. We also obtain a polynomial rate of convergence for the analogue of the quantum ergodicity theorem in the case of the rational torus.