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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.

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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.

EDIT: Cases where a positive integer can be represented in many ways as a sum of squares present even more multiplicity.

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