2023/01/09 by Hans Christianson, Christianson, Hans, Daniel Pezzi +1
Mathematics · Physics and Astronomy · #35G15 #35P05 #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Graph theory and applications #Quantum Mechanics and Applications #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2301.03555
openalex publication_date 2023/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we continue the study of eigenfunctions on triangles initiated by the first author in \citeChr-tri and \citeChr-simp. The Neumann data of Dirichlet eigenfunctions on triangles enjoys an equidistribution law, being equidistributed on each side. The proof of this result is remarkably simple, using only the radial vector field and a Rellich type integrations by parts. The equidistribution law, including on higher dimensional simplices, agrees with what Quantum Ergodic Restriction would predict. However, distribution of the Neumann data on subsets of a side is not well understood, and elementary methods do not appear to give enough information to draw conclusions. In the present note, we first show that an "obvious" conjecture fails even for the simplest right isosceles triangle using only Fourier series. We then use a result of Marklof-Rudnick \citeMarklof-Rudnick in which the authors show an interior \it spatial equidistribution law for a density-one subsequence of eigenfunctions to give an estimate on energy distribution of eigenfunctions on the interior. Finally we present some numerical computations suggesting the behaviour of eigenfunctions on almost isosceles triangles is quite complicated.