Nodal domains of a non-separable problem - the right angled isosceles triangle

TL;DR

This paper investigates the nodal patterns of Laplace eigenfunctions on a right-angled isosceles triangle, providing algorithms, recursive formulas, and statistical analysis revealing complex structures and links to classical dynamics.

Contribution

It introduces a new recursive formula for counting nodal domains and analyzes their distribution, uncovering richer structures than in separable or irregular shapes.

Findings

Recursive formula accurately predicts nodal domain counts.

Nodal count distribution exhibits complex, rich structure.

Links between nodal counts and classical periodic orbits are demonstrated.

Abstract

We study the nodal set of eigenfunctions of the Laplace operator on the right angled isosceles triangle. A local analysis of the nodal pattern provides an algorithm for computing the number of nodal domains for any eigenfunction. In addition, an exact recursive formula for the number of nodal domains is found to reproduce all existing data. Eventually we use the recursion formula to analyse a large sequence of nodal counts statistically. Our analysis shows that the distribution of nodal counts for this triangular shape has a much richer structure than the known cases of regular separable shapes or completely irregular shapes. Furthermore we demonstrate that the nodal count sequence contains information about the periodic orbits of the corresponding classical ray dynamics.