Points on nodal lines with given direction

TL;DR

This paper investigates the directional distribution of nodal lines of Laplacian eigenfunctions, providing bounds on the flat torus and calculating expected values for arithmetic random waves.

Contribution

It introduces new bounds for nodal line points on the flat torus and computes expected directional counts for arithmetic random waves.

Findings

Upper bounds for nodal points on the flat torus

Expected number of directional points for arithmetic random waves

Insights into the geometry of nodal lines

Abstract

We study of the directional distribution function of nodal lines for eigenfunctions of the Laplacian on a planar domain. This quantity counts the number of points where the normal to the nodal line points in a given direction. We give upper bounds for the flat torus, and compute the expected number for arithmetic random waves.