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

Let $-λ_j$ be the eigenvalues of the Laplace operator on the unit disk with Dirichlet conditions. The distribution $h(t) = \sum_j e^{i\sqrtλ_j t}$ 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, $t_k < 2π$, the perimeter of the circle, and $t_k$ tends to $2π$. In this paper, we show that $h(t)$ is infinitely differentiable as $t$ tends to $2π$ from the right.