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

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.

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Cited by in corpus (9)

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• The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree
• Nodal domains of the equilateral triangle billiard
• Exact eigenfunction amplitude distributions of integrable quantum billiards
• A nodal domain theorem for integrable billiards in two dimensions
• Courant-sharp eigenvalues of Neumann 2-rep-tiles
• Raising and lowering operators for quantum billiards
• On the Nodal Count Statistics for Separable Systems in any Dimension