Sharp bounds for the intersection of nodal lines with certain curves

Abstract

Let $Y$ be a hyperbolic surface and let $ϕ$ be a Laplacian eigenfunction having eigenvalue $-1/4-τ^2$ with $τ>0$. Let $N(ϕ)$ be the set of nodal lines of $ϕ$. For a fixed analytic curve $γ$ of finite length, we study the number of intersections between $N(ϕ)$ and $γ$ in terms of $τ$. When $Y$ is compact and $γ$ a geodesic circle, or when $Y$ has finite volume and $γ$ is a closed horocycle, we prove that $γ$ is "good" in the sense of [TZ]. As a result, we obtain that the number of intersections between $N(ϕ)$ and $γ$ is $O(τ)$. This bound is sharp.