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 1/4+r_j^2 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.