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Eigenfunction scarring and improvements in \(L^\infty\) bounds. (English) Zbl 1386.35307

Summary: 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.

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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