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 $$\frac{1}{4} + r_j^2 \to \infty $$
 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.