Log-scale equidistribution of nodal sets in Grauert tubes

Abstract Let M τ 0 be the Grauert tube (of some fixed radius τ 0 ) of a compact, negatively curved, real analytic Riemannian manifold M without boundary. Let φ λ be a Laplacian eigenfunction on M of eigenvalue − λ 2 and let φ λ C be its holomorphic extension to M τ 0 . In this article, we prove that on M τ 0 ∖ M , there exists a dimensional constant α > 0 and a full density subsequence { λ j k } k = 1 ∞ of the spectrum for which the masses of the complexified eigenfunctions φ λ j k C are asymptotically equidistributed at length scale ( log ⁡ λ j k ) − α . Moreover, the complex zeros of φ λ j k C also become equidistributed on this logarithmic length scale.