Vol. 11, No. 3, 2018

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Eigenfunction scarring and improvements in $L^\infty$ bounds

Jeffrey Galkowski and John A. Toth

Vol. 11 (2018), No. 3, 801–812
Abstract

We study the relationship between L growth of eigenfunctions and their L2 concentration as measured by defect measures. In particular, we show that scarring in the sense of concentration of defect measure on certain submanifolds is incompatible with maximal L growth. In addition, we show that a defect measure which is too diffuse, such as the Liouville measure, is also incompatible with maximal eigenfunction growth.

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Keywords
eigenfunction, sup norms, defect measure
Mathematical Subject Classification 2010
Primary: 35P20, 58J50
Milestones
Received: 4 April 2017
Revised: 18 August 2017
Accepted: 16 October 2017
Published: 22 November 2017
Authors
Jeffrey Galkowski
Department of Mathematics and Statistics
McGill University
Montréal, QC
Canada
John A. Toth
Department of Mathematics and Statistics
McGill University
Montréal, QC
Canada