Eigenfunction Concentration for Polygonal Billiards

In this note, we extend the results on eigenfunction concentration in billiards as proved by the third author in [8 Marzuola , J. ( 2007 ). Eigenfunctions for partially rectangular billiards . Communications in Partial Differential Equations 31 : 775 – 790 .[Taylor & Francis Online] , [Google Scholar]]. There, the methods developed in Burq and Zworski [3 Burq , N. , Zworski , M. ( 2005 ). Bouncing ball modes and quantum chaos . SIAM Review 47 : 43 – 49 .[Crossref], [Web of Science ®] , [Google Scholar]] 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.