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An Isaac Newton Institute Workshop

Random Matrix Theory and Arithmetic Aspects of Quantum Chaos

Complex zeros of real ergodic eigenfunctions

Author: Steve Zelditch (Johns Hopkins)

Abstract

A well-known problem in geometry of eigenfunctions of Laplacians on Riemannian manifolds is to determine how the nodal hypersurface (zero set) is asymptotically distributed as the eigenvalue tends to infinity. The random wave model predicts that the normalized measure of integration over the nodal hypersurface tends to the volume measure on the manifold. My talk is a preliminary report on the distribution of complex zeros of analytic continuations of eigenfunctions of real analytic Riemannian manifolds with ergodic geodesic flow. We describe how the complex nodal hypersurfaces are distributed in the cotangent bundle. The (perhaps surprising) result is that the complex zeros concentrate around the real ones.