Complex zeros of real ergodic eigenfunctions

Abstract

We determine the limit distribution (as λ→∞) of complex zeros for holomorphic continuations φλ ℂ to Grauert tubes of real eigenfunctions of the Laplacian on a real analytic compact Riemannian manifold (M,g) with ergodic geodesic flow. If $\{\phi_{j_{k}}\}$ is an ergodic sequence of eigenfunctions, we prove the weak limit formula $\frac{1}{\lambda_j}[Z_{\phi_{j_k}}^{\mathbb{C}}]\ \to\ \frac{i}{\pi} \partial\bar{\partial} |\xi|_g$ , where $[Z_{\phi_{j_k}^{\mathbb{C}}}]$ is the current of integration over the complex zeros and where $\overline{\partial}$ is with respect to the adapted complex structure of Lempert-Szöke and Guillemin-Stenzel.