Unconditional growth of nodal domains in a fixed compact set

Determine whether, for Maass cusp forms on the modular surface Γ\H and any fixed compact domain Ω⊂Γ\H, the number of nodal domains N^Ω(φ) tends to infinity as the Laplace eigenvalue λ_φ→∞, without any unproven hypotheses.

Background

The paper reviews the Bogomolny–Schmit prediction that the number of nodal domains of a Maass cusp form in a compact region grows linearly with the Laplace eigenvalue. Despite numerical support, this remains beyond current techniques. The authors emphasize that even the much weaker statement that the number of nodal domains in a fixed compact set tends to infinity with the eigenvalue is not known unconditionally.

This problem is central to understanding nodal domain statistics on arithmetic hyperbolic surfaces and connects to questions in quantum chaos. The authors focus on sign changes along geodesics as a proxy for counting nodal domains, but the unconditional global growth of nodal domains in compact sets remains unresolved.