Complex zeros of real ergodic eigenfunctions

Abstract

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