Delocalization of quasimodes on the disk

Abstract

This note deals with semiclassical measures associated to {(sufficiently accurate)} quasimodes $(u_h)$ for the Laplace-Dirichlet operator on the disk. In this time-independent set-up, we simplify the statements of our preprint arXiv:1406.0681 and their proofs. We describe the restriction of semiclassical measures to every invariant torus in terms of two-microlocal measures. As corollaries, we show regularity and delocalization properties for limit measures of $|u_h|^2 dx$: these are absolutely continuous in the interior of the disk and charge every open set intersecting the boundary.