Partially Localized Quasimodes in Large Subspaces

TL;DR

This paper demonstrates that in large subspaces of high-energy quasimodes on hyperbolic surfaces, one can find specific quasimodes that localize a significant portion of their mass on a singular set, revealing partial localization phenomena.

Contribution

It establishes the existence of localized quasimodes within large subspaces, extending understanding of quantum ergodicity and localization in hyperbolic geometry.

Findings

Existence of localized quasimodes in large subspaces

Localization on a codimension 1 singular set

Sharpness of results relative to previous QUE work

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.