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

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(λ) 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(λ). 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(1) improvements for manifolds with ergodic geodesic flows. Our results work for a full density subsequence of any given orthonormal basis of eigenfunctions.

eigenfunctions, nodal sets, doubling estimates, order of vanishing, quantum ergodicity
Mathematical Subject Classification 2010
Primary: 35P20
Received: 1 September 2016
Revised: 16 July 2017
Accepted: 28 September 2017
Published: 12 January 2018
Hamid Hezari
Department of Mathematics
University of California at Irvine
Irvine, CA
United States