Complex zeros of real ergodic eigenfunctions

摘要

We determine the limit distribution (as lambda -> infinity) of complex zeros for holomorphic continuations p(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 {p(jk)} is an ergodic sequence of eigenfunctions, we prove the weak limit formula 1/lambda j [Z(pjk)(C)] -> i/pi partial derivative partial derivative vertical bar xi vertical bar(g), where [Z(pjk)(C)] is the current of integration over the complex zeros and where. is with respect to the adapted complex structure of Lempert-Szoke and Guillemin-Stenzel.