Vol. 11, No. 3, 2018

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
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 ${L}^{\infty }$ 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