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 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 . 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 improvements for manifolds with ergodic geodesic flows. Our results work for a full density subsequence of any given orthonormal basis of eigenfunctions.
Citation
Hamid Hezari. "Applications of small-scale quantum ergodicity in nodal sets." Anal. PDE 11 (4) 855 - 871, 2018. https://doi.org/10.2140/apde.2018.11.855
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