Nodal intersections for random waves against a segment on the 3-dimensional torus

TL;DR

This paper studies the number of intersections between random eigenfunctions on a 3D torus and a line segment, revealing universal proportionality in expectation and linking variance bounds to lattice point theory.

Contribution

It provides a universal formula for expected nodal intersections and establishes a novel connection between variance bounds and lattice point arithmetic properties.

Findings

Expected intersection number proportional to length and wavenumber

Variance bounds depend on arithmetic properties of the line

Links between nodal intersections and lattice points on spheres

Abstract

We consider random Gaussian eigenfunctions of the Laplacian on the three-dimensional flat torus, and investigate the number of nodal intersections against a straight line segment. The expected intersection number, against any smooth curve, is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. We found an upper bound for the nodal intersections variance, depending on the arithmetic properties of the straight line. The considerations made establish a close relation between this problem and the theory of lattice points on spheres.