Eigenfunction concentration for polygonal billiards

Abstract

In this note, we extend the results on eigenfunction concentration in billiards as proved by the third author in \cite{M1}. There, the methods developed in Burq-Zworski \cite{BZ3} to study eigenfunctions for billiards which have rectangular components were applied. Here we take an arbitrary polygonal billiard $B$ and show that eigenfunction mass cannot concentrate away from the vertices; in other words, given any neighbourhood $U$ of the vertices, there is a lower bound $${\int }_U|u{|}^2\ge c{\int }_B|u{|}^2$$ for some $c=c(U)>0$ and any eigenfunction $u$.

9 pages, 3 figures