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Let $-\lambda_j$ be the eigenvalues of the Laplace operator on the unit diskwith Dirichlet conditions. The distribution $h(t) = \sum_j e^{i\sqrt\lambda_jt}$ is the trace of the solution operator of the wave equation on the disk. Itis well known that $h$ has isolated singularities at the lengths of thereflecting geodesics. In particular, $h$ is singular at $t_k$, the perimeter ofthe regular inscribed polygon with $k$ sides. Evidently, $t_k < 2\pi$, theperimeter 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.

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