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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 $$ n(\lambda,H):= \# (H \capN_{\phi_{\lambda}} ).$$ We first prove that for general piecewise-analyticdomains, and under an appropriate "goodness" condition on $H$, $$ n(\lambda,H)= {\mathcal O}_H(\lambda) (*)$$ as $\lambda \rightarrow \infty.$ We then provethat the bound in $(*)$ is satisfied in the case of quantum ergodic (QE)sequences of interior eigenfunctions, provided $\Omega$ is convex and $H$ hasstrictly positive geodesic curvature.

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