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

40 pages, 1 figure