# Simplicity of eigenvalues of an elliptic operator under a symmetry assumption

A striking difference in the spectral analysis of 2nd order elliptic boundary-value problems between one and several space dimensions is the following. In one space dimension, the eigenvalues are simple (Sturm-Liouville theory) though they can be multiple in several space dimensions. For instance the eigenfunctions of the Laplacian in the disk $D_2\subset{\mathbb R}^2$ have the form $f(r)\cos n(\theta-\phi)$ in polar coordinates; if $n\ge1$, then the corresponding eigenvalue has double multiplicity.

Let me assume that the data (domain $\Omega\subset{\mathbb R}^2$, operator, boundary conditions) be invariant under a symmetry, say $x_1\leftrightarrow-x_1$. Then one may consider the restriction of the boundary-value problem to either the space $E_+$ of even functions ($u(x)=u(-x_1,x_2)$) or the space $E_-$ of odd functions ($u(x)=-u(-x_1,x_2)$). The full spectrum is then the union of the even and odd spectra, with addition of multiplicities. Here is an example of such a problem $$\Omega=D_2,\qquad \Delta u=\lambda u\quad\hbox{ in }D_2,\qquad\frac{\partial u}{\partial r}=\rho(\theta)u\quad\hbox{ on }r=1,$$ where $\rho$ is an even function. For an other example, one may replace the boundary condition by $$\frac{\partial u}{\partial r}=\rho(\theta)\frac{\partial u}{\partial \theta}.$$

Is it true or not that the eigenvalues of each restricted problem (even or odd) are all simple ?

At least, the Krein-Rutman theory tells us that the first eigenvalue is simple, hence the first even eigenvalue is simple.

• Dear @Wolfgang, please do not edit more than three old questions each day. Editing many old threads in quick succession floods the front page and is strongly discouraged by the community. Please read the discussion at meta.mathoverflow.net/questions/599/… to learn about this limitation and some related past events. Thank you. – Ricardo Andrade Oct 11 '14 at 14:39
• @RicardoAndrade yes sorry again, I wasn't aware of that. I have already replied to the moderators as well. Will do better in the future! – Wolfgang Oct 11 '14 at 14:51

For example, consider the Laplacian on $[-1,1] \times [-1,1]$ with Dirichlet boundary conditions. If $m$ and $n$ are distinct positive integers, $\sin(m \pi x) \sin(n \pi y)$ and $\sin(n \pi x) \sin(m \pi y)$ are both eigenfunctions for the same eigenvalue, and are both odd. Similarly $\cos((m+1/2) \pi x) \cos((n+1/2) \pi y)$ and $\cos((n+1/2) \pi x) \cos((m+1/2) \pi y)$ which are both even.