Eigenfunction concentration for polygonal billiards

TL;DR

This paper proves that in polygonal billiards, eigenfunctions cannot concentrate away from vertices, ensuring a minimum mass near vertices regardless of eigenfunction choice.

Contribution

It extends previous eigenfunction concentration results from rectangular billiards to arbitrary polygonal billiards, establishing lower bounds near vertices.

Findings

Eigenfunction mass cannot concentrate away from vertices.

A positive lower bound exists for eigenfunction mass near vertices.

Results apply to any eigenfunction in 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 $$ \int_U |u|^2 \geq c \int_B |u|^2 $$ for some $c = c(U) > 0$ and any eigenfunction $u$.