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

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

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