Eigenfunction scarring and improvements in L∞ bounds

Galkowski, Jeffrey; Toth, John A.

doi:10.2140/apde.2018.11.801

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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

DOI: 10.2140/apde.2018.11.801

Abstract

We study the relationship between $L^{\infty}$ growth of eigenfunctions and their $L^{2}$ 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 $L^{\infty}$ 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.

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

Vol. 11, No. 3, 2018