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 phi(jk) on a Riemannian manifold (M, g) for which the number N(phi(jk)) of nodal domains tends to infinity with the eigenvalue. Our main result is that N(phi(jk))->+infinity along a subsequence of eigenvalues of density 1 if (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