On nodal sets and nodal domains on S^2 and R^2

TL;DR

This paper explores the topological structures of nodal sets for spherical harmonics on S^2 and constructs a solution to a specific PDE in R^2 with only two nodal domains, relevant to high energy eigenfunctions.

Contribution

It provides new insights into the topology of nodal sets on S^2 and presents a novel solution to the PDE Delta u = u in R^2 with minimal nodal domains.

Findings

Topological configurations of nodal sets on S^2 analyzed

Constructed a solution with only two nodal domains in R^2

Insights relevant to high energy eigenfunctions

Abstract

We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on S^2. We also construct a solution of the equation Delta u=u in R^2 that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.