A nodal domain theorem for integrable billiards in two dimensions

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 Schrödinger 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 $m\mathrm{mod}kn$, 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.

13 pages, 5 figures

References in corpus (1)

• Nodal domains of the equilateral triangle billiard

Cited by in corpus (5)

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• Raising and lowering operators for quantum billiards
• Quantum error correction beyond the toric code: dynamical systems meet encoding