On the behavior of nodal lines near the boundary for Laplace eigenfunctions on the square

TL;DR

This paper investigates how Dirichlet boundary conditions influence the expected length of nodal lines of random Laplace eigenfunctions on a square, revealing boundary effects are negligible at large scales.

Contribution

It provides a two-term asymptotic expansion for the expected nodal length near the boundary, independent of position, using new lattice point analysis.

Findings

Expected nodal length asymptotics are uniform across the square.

Boundary effects are negligible at scales larger than the Planck scale.

The asymptotic expansion holds along a density one sequence of energy levels.

Abstract

We are interested in the effect of Dirichlet boundary conditions on the nodal length of Laplace eigenfunctions. We study random Gaussian Laplace eigenfunctions on the two dimensional square and find a two terms asymptotic expansion for the expectation of the nodal length in any square of side larger than the Planck scale, along a denisty one sequence of energy levels. The proof relies on a new study of lattice points in small arcs, and shows that the said expectation is independent of the position of the square, giving the same asymptotic expansion both near and far from the boundaries.