It is an open problem in general to prove that there exists a sequence of (Formula presented.) -eigenfunctions (Formula presented.) on a Riemannian manifold (M, g) for which the number (Formula presented.) of nodal domains tends to infinity with the eigenvalue. Our main result is that (Formula presented.) 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.
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