A nodal domain theorem for integrable billiards in two dimensions

TL;DR

This paper establishes a mathematical link between integrable billiards and difference equations by analyzing the nodal domains of eigenfunctions, revealing pattern classifications and algebraic representations for both separable and non-separable cases.

Contribution

It introduces a novel difference equation framework for understanding nodal domain counts in all integrable billiards, including non-separable cases, and classifies eigenfunctions into families based on quantum numbers.

Findings

Nodal domain counts satisfy specific difference equations.

Eigenfunctions can be grouped into families labeled by quantum number classes.

Algebraic representations of nodal patterns are identified and explained.

Abstract

Eigenfunctions of integrable planar billiards are studied - in particular, the number of nodal domains, $\nu$, of the eigenfunctions are considered. The billiards for which the time-independent Schr\"odinger equation (Helmholtz equation) is separable admit trivial expressions for the number of domains. Here, we discover that for all separable and non-separable integrable billiards, $\nu $ satisfies certain difference equations. This has been possible because the eigenfunctions can be classified in families labelled by the same value of $m\mod kn$, given a particular $k$, for a set of quantum numbers, $m, n$. Further, we observe that the patterns in a family are similar and the algebraic representation of the geometrical nodal patterns is found. Instances of this representation are explained in detail to understand the beauty of the patterns. This paper therefore presents a mathematical connection between integrable systems and difference equations.