Ergodicity and intersections of nodal sets and geodesics on real analytic surfaces

Abstract

We consider the the intersections of the complex nodal set of the analytic continuation of an eigenfunction of the Laplacian on a real analytic surface with the complexification of a geodesic. We prove that if the geodesic flow is ergodic and if the geodesic is periodic and satisfies a generic asymmetry condition, then the intersection points condense along the real geodesic and become uniformly distributed with respect to its arc-length. We prove an analogous result for non-periodic geodesics except that the `origin' is allowed to move with $\lambda_j$. The proof uses the quantum ergodic restriction theorem due to J. Toth and the author (see also Dyatlov-Zworski).