Vol. 11, No. 4, 2018

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Applications of small-scale quantum ergodicity in nodal sets

Hamid Hezari

Vol. 11 (2018), No. 4, 855–871
Abstract

The goal of this article is to draw new applications of small-scale quantum ergodicity in nodal sets of eigenfunctions. We show that if quantum ergodicity holds on balls of shrinking radius $r\left(\lambda \right)\to 0$ then one can achieve improvements on the recent upper bounds of Logunov (2016) and Logunov and Malinnikova (2016) on the size of nodal sets, according to a certain power of $r\left(\lambda \right)$. We also show that the doubling estimates and the order-of-vanishing results of Donnelly and Fefferman (1988, 1990) can be improved. Due to results of Han (2015) and Hezari and Rivière (2016), small-scale QE holds on negatively curved manifolds at logarithmically shrinking rates, and thus we get logarithmic improvements on such manifolds for the above measurements of eigenfunctions. We also get $o\left(1\right)$ improvements for manifolds with ergodic geodesic flows. Our results work for a full density subsequence of any given orthonormal basis of eigenfunctions.

Keywords
eigenfunctions, nodal sets, doubling estimates, order of vanishing, quantum ergodicity
Primary: 35P20
Milestones
Received: 1 September 2016
Revised: 16 July 2017
Accepted: 28 September 2017
Published: 12 January 2018
Authors
 Hamid Hezari Department of Mathematics University of California at Irvine Irvine, CA United States