Ruled nodal surfaces of Laplace eigenfunctions and injectivity sets for the spherical mean Radon transform in $\mathbb R^3.$

TL;DR

This paper characterizes the structure of nodal sets of Laplace eigenfunctions in three dimensions, showing they are unions of cones or Coxeter systems, and confirms a conjecture relating to injectivity sets for the spherical mean transform.

Contribution

It provides a complete description of ruled nodal surfaces of Laplace eigenfunctions in R^3 and verifies a conjecture on ruled injectivity sets for the spherical mean Radon transform.

Findings

Nodal sets are unions of cones contained in zero sets of harmonic polynomials.

If the surface is a C^1 manifold, it forms a Coxeter system of planes.

The results confirm a conjecture of Quinto and the author regarding injectivity sets.

Abstract

It is proved that if a Paley-Wiener family of eigenfunctions of the Laplace operator in $\mathbb R^3$ vanishes on a real analytically ruled two-dimensional surface $S \subset \mathbb R^3$ then $S$ is a union of cones, each of which is contained in a translate of the zero set of a nonzero harmonic homogeneous polynomial. If $S$ is an immersed $C^1-$ manifold then $S$ is a Coxeter system of planes. Full description of common nodal sets of the Laplace spectra of convexly supported distributions is given. In equivalent terms, the result describes ruled injectivity sets for the spherical mean transform and confirms, for the case of ruled surfaces in $\mathbb R^3,$ a conjecture of E.T. Quinto and the author .