Delocalization of quasimodes on the disk

TL;DR

This paper investigates the behavior of semiclassical measures associated with quasimodes of the Laplace-Dirichlet operator on a disk, revealing their regularity and delocalization properties, especially near the boundary.

Contribution

It simplifies previous results on semiclassical measures for disk quasimodes and characterizes their restriction to invariant tori using two-microlocal measures.

Findings

Semiclassical measures are absolutely continuous inside the disk.

Limit measures charge every open set intersecting the boundary.

The paper provides a simplified proof of earlier results.

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.