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

TL;DR

This paper establishes bounds on the number of intersections between interior analytic curves and nodal sets of planar Neumann eigenfunctions, with sharper results for quantum ergodic sequences in convex domains.

Contribution

It proves a general linear bound on intersection counts for piecewise-analytic domains and verifies this bound for quantum ergodic eigenfunctions in convex domains with curved interior curves.

Findings

Intersection number grows at most linearly with eigenvalue parameter

Bound holds for general piecewise-analytic domains under certain conditions

Stronger bounds are confirmed for quantum ergodic eigenfunctions in convex domains

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.