Log-scale equidistribution of nodal sets in Grauert tubes

Abstract

Let $M_{\tau_0}$ be the Grauert tube (of some fixed radius $\tau_0$) of a compact, negatively curved, real analytic Riemannian manifold $M$ without boundary. Let $\phi_\lambda$ be a Laplacian eigenfunction on $M$ of eigenvalues $-\lambda^2$ and let $\phi_\lambda^\mathbb{C}$ be its holomorphic extension to $M_{\tau_0}$. In this article, we prove that on $M_{\tau_0} \setminus M$, there exists a dimensional constant $\alpha > 0$ and a full density subsequence $ \{\lambda_{j_k}\}_{k=1}^{\infty}$ of the spectrum for which the masses of the complexified eigenfunctions $\phi_{\lambda_{j_k}}^\mathbb{C}$ are asymptotically equidistributed at length scale $(\log \lambda_{j_k})^{-\alpha}$. Moreover, the complex zeros of $\phi_{\lambda_{j_k}}^\mathbb{C}$ also become equidistributed on this logarithmic length scale.