Log-scale equidistribution of nodal sets in Grauert tubes

Abstract

Let Mτ0 be the Grauert tube (of some fixed radius τ0) of a compact, negatively curved, real analytic Riemannian manifold M without boundary. Let φλ be a Laplacian eigenfunction on M of eigenvalue −λ2 and let φλC be its holomorphic extension to Mτ0. In this article, we prove that on Mτ0∖M, there exists a dimensional constant α>0 and a full density subsequence {λjk}k=1∞ of the spectrum for which the masses of the complexified eigenfunctions φλjkC are asymptotically equidistributed at length scale (log⁡λjk)−α. Moreover, the complex zeros of φλjkC also become equidistributed on this logarithmic length scale.