= Solution
The <five-point Laplacian> on the square grid is
$$
\boxed{\Delta_hW_{i,j}=\frac{W_{i+1,j}+W_{i-1,j}+W_{i,j+1}+W_{i,j-1}-4W_{i,j}}{h^2},\qquad1\leq i,j\leq N-1.}
$$
It adds the <second-order central differences> in the two coordinate directions and uses the given zero <Dirichlet boundary conditions> when a neighbouring node lies on the boundary.
The <sine addition formula> implies
$$
\sin((i+1)\alpha)+\sin((i-1)\alpha)-2\sin(i\alpha)
=2(\cos\alpha-1)\sin(i\alpha)
=-4\sin^2(\alpha/2)\sin(i\alpha).
$$
Apply this identity with $\alpha=m\pi h$ in the $x$ direction and with $\alpha=n\pi h$ in the $y$ direction. For $U_{i,j}^{(m,n)}=\sin(m\pi ih)\sin(n\pi jh)$, it gives
$$
-\Delta_hU_{i,j}^{(m,n)}+U_{i,j}^{(m,n)}
=\left[1+\frac4{h^2}\left(\sin^2\frac{m\pi h}2+\sin^2\frac{n\pi h}2\right)\right]U_{i,j}^{(m,n)}.
$$
Hence the discrete <eigenvalues> are
$$
\boxed{\Lambda_{m,n}=1+\frac4{h^2}\left(\sin^2\frac{m\pi h}2+\sin^2\frac{n\pi h}2\right)>0,\qquad1\leq m,n\leq N-1.}
$$
Both <sine> factors vanish on their respective boundary sides. Also $U_{1,1}^{(m,n)}>0$, since $m\pi/N$ and $n\pi/N$ lie strictly between zero and $\pi$. Thus each array is a nonzero discrete <eigenvector>, rather than merely the zero solution of the homogeneous system. The separated structure is also described by the <five-point Dirichlet Laplacian as a Kronecker sum>.
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