= Dirichlet eigensystem of a rectangle
{c}
{title2=$\lambda_{mn}=(m\pi/L_x)^2+(n\pi/L_y)^2$}
On $0<x<L_x$, $0<y<L_y$ with positive side lengths and zero <Dirichlet boundary conditions>, <separation of variables> for $-\Delta u=\lambda u$ gives
$$
u_{mn}(x,y)=\sin(m\pi x/L_x)\sin(n\pi y/L_y),\qquad m,n\geq1,
$$
with the displayed <eigenvalues>. The one-dimensional endpoint conditions exclude constant and hyperbolic factors and force the positive integer sine frequencies. The product <Fourier sine series> basis is complete in the square-integrable functions on the rectangle. Consequently an <eigenfunction> with prescribed $\lambda$ is a <linear combination> of exactly those modes with $\lambda_{mn}=\lambda$; if none match, only the zero solution exists. Distinct pairs with the same sum give an <eigenvalue> with <geometric multiplicity> greater than one. On a square of side $\pi$, for instance, $\lambda=25$ has precisely the pairs $(3,4)$ and $(4,3)$, so its <eigenspace> has <dimension> two.
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