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 [8]. There, the methods developed in Burq and Zworski [3] 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 neighborhood U of the vertices, there is a lower bound, for some c=c(U)>0 and any eigenfunction u.