Intersection bounds for nodal sets of planar Neumann eigenfunctions with interior analytic curves

Abstract

Let \( \Omega \subset R^2\) be a bounded piecewise smooth domain and \(\phi_\lambda\) be a Neumann (or Dirichlet) eigenfunction with eigenvalue \(\lambda^2\) and nodal set \({ N}_{\phi_{\lambda}} = {x \in \Omega; \phi_{\lambda}(x) = 0}.\) Let \(H \subset \Omega\) be an interior \(C^{\omega}\) curve. Consider the intersection number \[ n(\lambda,H):= \# (H \cap N_{\phi_{\lambda}} ).\] We first prove that for general piecewise-analytic domains, and under an appropriate "goodness" condition on \(H\), \[ n(\lambda,H) = {\mathcal O}_H(\lambda) (*)\] as \(\lambda \rightarrow \infty.\) We then prove that the bound in \((*)\) is satisfied in the case of quantum ergodic (QE) sequences of interior eigenfunctions, provided \(\Omega\) is convex and \(H\) has strictly positive geodesic curvature.