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

2010/12/30 by Yves Colin de Verdìère, Yves Colin de Verdière, de Verdière, Yves Colin +4
Mathematics · #35P20 #35R30 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Mathematical functions and polynomials #Spectral Theory in Mathematical Physics #math.AP #msc:35P20 #msc:35R30

paper · pdf · doi:10.48550/arxiv.1101.0099

24 pages

arxiv created 2010/12/30 · openalex publication_date 2010/12/30 · arxiv updated 2011/01/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let -λ_j be the eigenvalues of the Laplace operator on the unit disk with Dirichlet conditions. The distribution h(t) = ∑_j e^{i√λ_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.

Citations

• The propagation of singularities along gliding rays

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