We investigate the singularities of the trace of the half-wave group, $mathrm{Tr} , e^{-itsqrtDelta}$, on Euclidean surfaces with conical singularities $(X,g)$. We compute the leading-order singularity associated to periodic orbits with successive de
generate diffractions. This result extends the previous work of the third author cite{Hil} and the two-dimensional case of the work of the first author and Wunsch cite{ForWun} as well as the seminal result of Duistermaat and Guillemin cite{DuiGui} in the smooth setting. As an intermediate step, we identify the wave propagators on $X$ as singular Fourier integral operators associated to intersecting Lagrangian submanifolds, originally developed by Melrose and Uhlmann cite{MelUhl}.
We study the problem of estimating the $L^2$ norm of Laplace eigenfunctions on a compact Riemannian manifold $M$ when restricted to a hypersurface $H$. We prove mass estimates for the restrictions of eigenfunctions $phi_h$, $(h^2 Delta - 1)phi_h = 0$
, to $H$ in the region exterior to the coball bundle of $H$, on $h^{delta}$-scales ($0leq delta < 2/3$). We use this estimate to obtain an $O(1)$ $L^2$-restriction bound for the Neumann data along $H.$ The estimate also applies to eigenfunctions of semiclassical Schrodinger operators.
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 $u$.
