= Dirichlet and Neumann spectra of a square
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{title2=$\lambda_{jk}=j^2+k^2\quad\text{on }[0,\pi]^2$}
For $-\Delta\Phi=\lambda\Phi$ on the square, <separation of variables> gives <Dirichlet eigenfunctions> $\sin(jx)\sin(ky)$ with $j,k\geq1$, and <Neumann eigenfunctions> $\cos(jx)\cos(ky)$ with $j,k\geq0$. The <eigenvalues> in both cases are $j^2+k^2$, with multiplicity equal to the number of ordered admissible pairs. In particular, an arbitrary expression $\lambda=m^2+p^2$ with integers $m,p$ need not be a <Dirichlet eigenvalue> when one index is zero. The Neumann zero eigenvalue has the constant eigenfunction, whereas the Dirichlet zero problem has only the zero solution by <Dirichlet uniqueness for the Poisson equation>.
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