Complex zeros of real ergodic eigenfunctions

Abstract

We determine the limit distribution (as $\lambda \to \infty$) 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 $\{{\varphi }_{j_k}\}$ 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.

Added some examples and references. Also added a new Corollary, and corrected some typos

References in corpus (2)

• Distribution of zeros of random and quantum chaotic sections of positive line bundles
• Quantum ergodicity of C* dynamical systems

Cited by in corpus (2)

• Lower bounds for nodal sets of eigenfunctions
• Algebras of pseudo-differential operators acting on holomorphic Sobolev spaces