Abstract.
This article reviews two rigorous results about the complex zeros of eigenfunctions of the Laplacian, that is, the zeros of the analytic continuation of the eigenfunctions to the complexification of the underlying space. Such a complexification of the problem is analogous to studying the complex zeros of polynomials with real coefficients. The first result determines the limit distribution of complex zeros of `ergodic eigenfunctions' such as eigenfunctions of classically chaotic systems. The second result determines the expected distribution of complex zeros for complexifications of Gaussian random waves adapted to the Riemannian manifold. The resulting distribution is the same in both cases. It is singular along the set of real points.
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Zelditch, S. Nodal lines, ergodicity and complex numbers. Eur. Phys. J. Spec. Top. 145, 271–286 (2007). https://doi.org/10.1140/epjst/e2007-00162-3
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DOI: https://doi.org/10.1140/epjst/e2007-00162-3