Eigenfunction concentration for polygonal billiards

Hassell, Andrew; Hillairet, Luc; Marzuola, Jeremy

Eigenfunction concentration for polygonal billiards

Authors

Hassell, Andrew

Hillairet, Luc

Marzuola, Jeremy

Description

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 $$ \int_U |u|^2 \geq c \int_B |u|^2 $$ for some $c = c(U) > 0$ and any eigenfunction $u$.
9 pages, 3 figures