Counting nodal domains on surfaces of revolution

TL;DR

This paper analyzes the distribution of nodal domains of eigenfunctions on surfaces of revolution, showing that the nodal sequence uniquely determines the surface's shape in the classical limit.

Contribution

It introduces a statistical analysis of nodal domain counts on surfaces of revolution and proves the nodal sequence uniquely identifies the surface shape.

Findings

Nodal domain counts form a predictable distribution approaching a limit.

The distribution of normalized nodal counts converges as spectrum index increases.

Nodal sequence uniquely determines the surface shape up to scaling.

Abstract

We consider eigenfunctions of the Laplace-Beltrami operator on special surfaces of revolution. For this separable system, the nodal domains of the (real) eigenfunctions form a checker-board pattern, and their number $\nu_n$ is proportional to the product of the angular and the "surface" quantum numbers. Arranging the wave functions by increasing values of the Laplace-Beltrami spectrum, we obtain the nodal sequence, whose statistical properties we study. In particular we investigate the distribution of the normalized counts $\frac{\nu_n}{n}$ for sequences of eigenfunctions with $K \le n\le K + \Delta K$ where $K,\Delta K \in \mathbb{N}$. We show that the distribution approaches a limit as $K,\Delta K\to\infty$ (the classical limit), and study the leading corrections in the semi-classical limit. With this information, we derive the central result of this work: the nodal sequence of a mirror-symmetric surface is sufficient to uniquely determine its shape (modulo scaling).