A difference-equation formalism for the nodal domains of separable billiards

TL;DR

This paper extends a difference-equation formalism for counting nodal domains to all integrable billiards, providing explicit solutions and broadening the understanding of their spectral properties.

Contribution

It generalizes a difference-equation approach to all integrable billiards, enabling explicit calculation of nodal domain counts.

Findings

Difference equations accurately describe nodal domain counts in integrable billiards

Explicit solutions for these equations are derived

The formalism applies to all separable billiard systems

Abstract

Recently, the nodal domain counts of planar, integrable billiards with Dirichlet boundary conditions were shown to satisfy certain difference equations in [Ann. Phys. 351, 1-12 (2014)]. The exact solutions of these equations give the number of domains explicitly. For complete generality, we demonstrate this novel formulation for three additional separable systems and thus extend the statement to all integrable billiards.