Partially Localized Quasimodes in Large Subspaces

2011/11/07 by Shimon Brooks, Brooks, Shimon
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Spectral Theory (math.SP) #math.DS #math.SP

paper · pdf · doi:10.48550/arxiv.1111.1626

18 pages

arxiv created 2011/11/07 · openalex publication_date 2011/11/07 · arxiv updated 2011/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 (cr_j)-dimensional spaces S_j of quasimodes, where 1/4+r_j² is an approximate eigenvalue for S_j. Then we can find a vector psi_j in each S_j, such that any weak-* limit point of the microlocal lifts of |psi_j|² localizes a positive proportion of its mass on a singular set of codimension 1. This result is sharp, in light of recent joint work with E. Lindenstrauss, proving QUE for certain joint quasimodes that include spaces of size o(r_j) with arbitrarily slow decay.

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