Eigenfunction concentration for pseudointegrable billiards

In this note, we extend the results on eigenfunction concentration in pseudointegrable billiards as proved by the second 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 pseudointegrable 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 $c > 0$ and any eigenfunction $u$. The results apply also to plane domains, or tori, with slits, with pseudointegrable billiard flow.