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.