In mathematics, the nodal line conjecture is a statement posed in 1967 by Lawrence E. Payne about the Laplacian partial differential equation. The original conjecture predicts that for the Dirichlet problem on a bounded two-dimensional domain, the second eigenfunction has a nodal line that meets the boundary of the domain. The general conjecture was proved false in 1997 by carving a large number of microscopic holes out of a disk, a technique that simulates a Schrödinger potential. Other positive and negative results are known for various special cases of domains; in general, it remains an open problem to describe how simple the domain must be for the conjecture to hold.
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