Partially localized quasimodes in large subspaces

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 $$\frac{1}{4} + r_j^2 \to \infty $$ is an approximate eigenvalue for S j . Then we can find a sequence of vectors ψ j ∈ S j , such that any weak-* limit point of the microlocal lifts of |ψ j |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(r j ), with arbitrarily slow decay.