Singularities of the wave trace near cluster points of the length spectrum

Abstract

Let $-{\lambda }_j$ be the eigenvalues of the Laplace operator on the unit disk with Dirichlet conditions. The distribution $h(t)={\sum }_j{e}^{i{\sqrt{\lambda }}_{j}t}$ is the trace of the solution operator of the wave equation on the disk. It is well known that $h$ has isolated singularities at the lengths of the reflecting geodesics. In particular, $h$ is singular at $t_k$, the perimeter of the regular inscribed polygon with $k$ sides. Evidently, $t_k<2\pi$, the perimeter of the circle, and $t_k$ tends to $2\pi$. In this paper, we show that $h(t)$ is infinitely differentiable as $t$ tends to $2\pi$ from the right.

24 pages