# Sharp bounds for the intersection of nodal lines with certain curves

### Junehyuk Jung

Princeton University, USA

## Abstract

Let $Y$ be a hyperbolic surface and let $\phi$ be a Laplacian eigenfunction having eigenvalue $-1/4-\tau^2$ with $\tau>0$. Let $N(\phi)$ be the set of nodal lines of $\phi$. For a fixed analytic curve $\gamma$ of finite length, we study the number of intersections between $N(\phi)$ 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 \cite{pa}. As a result, we obtain that the number of intersections between $N(\phi)$ and $\gamma$ is $O(\tau)$. This bound is sharp.

## Cite this article

Junehyuk Jung, Sharp bounds for the intersection of nodal lines with certain curves. J. Eur. Math. Soc. 16 (2014), no. 2, pp. 273–288

DOI 10.4171/JEMS/433