Counting nodal domains

Animations from a recent work

by Bernard Helffer and Mikael Persson Sundqvist

The problem

A classical problem in analysis consists in counting the number of nodal domains of eigenfunctions to the Laplace operator. The Courant theorem states that the $n$th eigenfunction (counting multiplicity) $\Psi_n$ of the operator $-\Delta$ in a bounded domain has at most $n$ nodal domains.

The purpose of this work was to find all possible $n$ such that $\Psi_n$ is Courant sharp, in the sense that it has exactly $n$ nodal domains, if the domain is a square and if Neumann boundary conditions are imposed. It turned out that only five eigenvalues have this property.

The Courant sharp cases

Here we show plots of the Courant sharp cases. In fact, all such cases are built from eigenfunctions on the form \[ \Phi_{p,q}^{\theta}(x,y)=\cos\theta\cos px\cos qy+\sin\theta\cos qx\cos py, \] where $\theta$ is a parameter, $x$ and $y$ our variables and $p^2+q^2=\lambda$ is the eigenvalue. Below we have plotted the set $$ \bigl\{(x,y)\in \mathbb{R}^2~|~0\lt x\lt \pi,\ 0\lt y\lt \pi,\ \Phi_{p,q}^{\theta}(x,y)\gt 0\bigr\}\tag{1} $$ in black and its complement in white.

Animations of nodal domains

In the cases where the eigenspace is two-dimensional, the set in equation $(1)$ above varies with $\theta$. Below we show how.

A preprint

You can read the full article on arXiv. It is accepted for publication in Moscow Mathematical Journal.