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

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.

10 pages

Cited by in corpus (3)

• On the growth of eigenfunction averages: microlocalization and geometry
• Growth of high $L^p$ norms for eigenfunctions: an application of geodesic beams
• Defect measures of eigenfunctions with maximal $L^{\infty }$ growth