Complex zeros of real ergodic eigenfunctions

Details Complex zeros of real ergodic eigenfunctions Complex zeros of real ergodic eigenfunctions

2005-05-24

arxiv.org/abs/math/0505513v4

Invent. Math. 167 (2007), no. 2, 419--443

Mathematics

Spectral Theory

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

Scientific paper

10.1007/s00222-006-0024-z

We determine the limit distribution (as $\lambda \to \infty$) of complex zeros for holomorphic continuations $\phi_{\lambda}^{\C}$ 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 $\frac{1}{\lambda_j} [Z_{\phi_{j_k}^{\C}}] \to \frac{i}{\pi} \bar{\partial} {\partial} |\xi|_g$, where $ [Z_{\phi_{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-Sz\"oke and Guillemin-Stenzel.

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