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

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

2010-12-30

arxiv.org/abs/1101.0099v1

Mathematics

Analysis of PDEs

24 pages

Scientific paper

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.

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