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 ${\varphi }_{\lambda }$ be a Laplacian eigenfunction on $M$ of eigenvalues $-{\lambda }^2$ and let ${\varphi }_{\lambda }^{\mathbb{C}}$ 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 ${\varphi }_{{\lambda }_{j_k}}^{\mathbb{C}}$ are asymptotically equidistributed at length scale $(\log {\lambda }_{j_k}{)}^{-\alpha }$. Moreover, the complex zeros of ${\varphi }_{{\lambda }_{j_k}}^{\mathbb{C}}$ also become equidistributed on this logarithmic length scale.