Nodal Set Openings on Perturbed Rectangular Domains

TL;DR

This paper investigates how small, smooth boundary perturbations of a rectangle affect the structure of eigenfunction nodal sets, revealing conditions under which crossings open into separated curves, with results largely independent of the rectangle's aspect ratio.

Contribution

It introduces an approximate separation of variables approach to analyze boundary perturbations' effects on eigenfunction nodal sets, highlighting aspect ratio independence.

Findings

Boundary perturbations can open nodal crossings into separated curves.

The nature of boundary perturbations influences the opening orientation and size.

Many features of the perturbed nodal set are asymptotically independent of aspect ratio.

Abstract

We study the effects of perturbing the boundary of a rectangle on the nodal sets of eigenfunctions of the Laplacian. Namely, for a rectangle of a given aspect ratio $N$, we identify the first Dirichlet mode to feature a crossing in its nodal set and perturb one of the sides of the rectangle by a close to flat, smooth curve. Such perturbations will often "open" the crossing in the nodal set, splitting it into two curves, and we study the separation between these curves and their regularity. The main technique used is an approximate separation of variables that allows us to restrict study to the first two Fourier modes in an eigenfunction expansion. We show how the nature of the boundary perturbation provides conditions on the orientation of the opening and estimates on its size. In particular, several features of the perturbed nodal set are asymptotically independent of the aspect ratio, which contrasts with prior works. Numerical results supporting our findings are also presented.