A non-concentration estimate for partially rectangular billiards

TL;DR

This paper establishes a lower bound on the localization of quasimodes in partially rectangular billiard domains, showing they cannot concentrate solely in the rectangular part, with implications for wave behavior in such geometries.

Contribution

It provides a non-concentration estimate for quasimodes in partially rectangular billiards, extending previous work to less smooth domains and improving bounds for smoother cases.

Findings

Quasimodes cannot concentrate entirely in the rectangular region.

Lower bounds on $L^2$ mass in the wings are established.

Results apply to domains with different smoothness levels.

Abstract

We consider quasimodes on planar domains with a partially rectangular boundary. We prove that for any $\epsilon_0>0$, an $\O(\lambda^{-\epsilon_0})$ quasimode must have $L^2$ mass in the "wings" bounded below by $\lambda^{-2-\delta}$ for any $\delta>0$. The proof uses the author's recent work on 0-Gevrey smooth domains to approximate quasimodes on $C^{1,1}$ domains. There is an improvement for $C^{2,\alpha}$ domains.