 Dirichlet and Neumann spectra of a square

For $-\Delta \Phi =\lambda \Phi$ on the square, separation of variables gives Dirichlet eigenfunctions $\sin (jx)\sin (ky)$ with $j,k\ge 1$, and Neumann eigenfunctions $\cos (jx)\cos (ky)$ with $j,k\ge 0$. 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.