Partially localized quasimodes in large subspaces

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 sequence of vectors ψ_j ∈ S_j, such that any weak-* limit point of the microlocal lifts of {pipe}ψ_j{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(r_j), with arbitrarily slow decay.