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

Abstract

Let $\Omega \subset \mathbb{R}^2$ be a bounded piecewise smooth domain and $\varphi \lambda$ be a Neumann (or Dirichlet) eigenfunction with eigenvalue $\lambda^2$ and nodal set $\mathcal{N}_{\varphi \lambda} = \{ x \in \Omega ; \varphi \lambda (x) = 0 \}$. Let $H \subset \Omega$ be an interior $C^{\omega}$ curve. Consider the intersection number \[ n(\lambda, H) := \# (H \cap \mathcal{N}_{\varphi \lambda}). \] We first prove that for general piecewise-analytic domains, and under an appropriate “goodness” condition on $H$ (see Theorem 1.1), \[ n(\lambda, H) = \mathcal{O}_H (\lambda) \] as $\lambda \to \infty$. Then, using Theorem 1.1, we prove in Theorem 1.2 that the bound in the above equation is satisfied in the case of quantum ergodic (QE) sequences of interior eigenfunctions, provided $\Omega$ is convex and $H$ has strictly positive geodesic curvature.