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

Abstract

It is an open problem in general to prove that there exists a sequence of ${\Delta }_g$-eigenfunctions ${\varphi }_{j_k}$ on a Riemannian manifold $(M,g)$ for which the number $N({\varphi }_{j_k})$ of nodal domains tends to infinity with the eigenvalue. Our main result is that $N({\varphi }_{j_k})\to \infty$ 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.

26 pages; we have revised and updated the article to take into the article arXiv 1411.1035 into account

References in corpus (1)

• Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary

Cited by in corpus (3)

• On the nodal lines of Eisenstein series on Schottky surfaces
• Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary
• Topological Properties of Neumann Domains