Partially localized quasimodes in large subspaces

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We consider spaces of high-energy quasimodes for the Laplacian on a compact hyperbolic surface, and show that when the spaces are large enough, one can find quasimodes that exhibit strong localization phenomena. Namely, take any constant c, and a sequence of crj-dimensional spaces Sj of quasimodes, where 1/4 + rj2 → ∞ is an approximate eigenvalue for Sj. Then we can find a sequence of vectors ψj ∈ Sj, such that any weak-* limit point of the microlocal lifts of {pipe}ψj{pipe}2 localizes a positive proportion of its mass on a singular set of codimension 1. This result is sharp, in light of the QUE result of [BL12] for certain joint quasimodes that include spaces of size o(rj), with arbitrarily slow decay.

Original languageEnglish
Pages (from-to)393-417
Number of pages25
JournalIsrael Journal of Mathematics
Issue number1
StatePublished - Nov 2013
Externally publishedYes

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