English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $M_{\tau_0}$ be the Grauert tube (of some fixed radius $\tau_0$) of acompact, negatively curved, real analytic Riemannian manifold $M$ withoutboundary. 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 $M_{\tau_0} \setminus M$, thereexists a dimensional constant $\alpha > 0$ and a full density subsequence $\{\lambda_{j_k}\}_{k=1}^{\infty}$ of the spectrum for which the masses of thecomplexified eigenfunctions $\phi_{\lambda_{j_k}}^\mathbb{C}$ areasymptotically equidistributed at length scale $(\log\lambda_{j_k})^{-\alpha}$. Moreover, the complex zeros of$\phi_{\lambda_{j_k}}^\mathbb{C}$ also become equidistributed on thislogarithmic length scale.

Log-scale equidistribution of nodal sets in Grauert tubes