# 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 $ϕ$ 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 \cite{pa}. As a result, we obtain that the number of intersections between $N(ϕ)$ and $γ$ is $O(τ)$. 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