Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary

TL;DR

This paper proves that on non-positively curved surfaces with concave boundary, there exists a subsequence of eigenfunctions whose number of nodal domains tends to infinity, advancing understanding of eigenfunction behavior in such geometries.

Contribution

It establishes that for non-positively curved surfaces with concave boundary, the nodal domain count diverges along a density 1 subsequence of eigenvalues, without requiring symmetries.

Findings

Nodal domain count tends to infinity along a subsequence of eigenvalues.

Results apply to non-positively curved surfaces with concave boundary.

No symmetry assumptions needed for the surfaces.

Abstract

It is an open problem in general to prove that there exists a sequence of $\Delta_g$-eigenfunctions $\phi_{j_k}$ on a Riemannian manifold $(M, g)$ for which the number $N(\phi_{j_k}) $ of nodal domains tends to infinity with the eigenvalue. Our main result is that $N(\phi_{j_k}) \to \infty$ along a subsequence of eigenvalues of density $1$ if the $(M, g)$ is a non-positively curved surface with concave boundary, i.e. a generalized Sinai or Lorentz billiard. Unlike the recent closely related work of Ghosh-Reznikov-Sarnak and of the authors on the nodal domain counting problem, the surfaces need not have any symmetries.