Sharp bounds for the intersection of nodal lines with certain curves

TL;DR

This paper establishes sharp bounds on the number of intersections between nodal lines of Laplacian eigenfunctions on hyperbolic surfaces and certain curves, showing the intersection count grows linearly with the eigenvalue parameter.

Contribution

It proves sharp linear bounds for intersections of nodal lines with geodesic circles and horocycles on hyperbolic surfaces, extending understanding of eigenfunction nodal geometry.

Findings

Number of intersections is O(τ) for eigenfunctions with eigenvalue -1/4 - τ^2.

Bounds are sharp, matching known examples.

Results apply to compact surfaces and finite volume surfaces with specific curves.

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 [TZ]. As a result, we obtain that the number of intersections between $N(\phi)$ and $\gamma$ is $O(\tau)$. This bound is sharp.