Energy Distribution for Dirichlet Eigenfunctions on Right Triangles

TL;DR

This paper studies how eigenfunctions' energy distributes on right triangles, proving equidistribution on sides, showing conjectures fail in simple cases, and exploring complex behaviors with numerical evidence.

Contribution

It proves equidistribution of Neumann data on triangle sides using simple methods, examines the failure of naive conjectures, and investigates complex eigenfunction behaviors through numerical analysis.

Findings

Neumann data of Dirichlet eigenfunctions is equidistributed on sides.

A simple Fourier series argument shows certain conjectures fail.

Numerical results indicate complex eigenfunction behaviors on nearly isosceles triangles.

Abstract

In this paper, we continue the study of eigenfunctions on triangles initiated by the first author in \cite{Chr-tri} and \cite{Chr-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 \cite{Marklof-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.