# 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.