Sharp bounds for the intersection of nodal lines with certain curves

Abstract

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