Discrete Helmholtz resonance (${\lambda }_{pq}=4h^{-2}[{\sin }^2(p\pi h/2)+{\sin }^2(q\pi h/2)]$)

On a square Dirichlet grid with spacing $h=1/J$ and interior indices $1,\dots ,J-1$, the five-point Laplacian is diagonalized by products of the sine modes used in the discrete sine transform. The discrete equation $({\Delta }_h+\lambda I)U=F$ is nonsingular exactly when $\lambda$ is not one of the displayed positive eigenvalues of $-{\Delta }_h$, with $1\le p,q\le J-1$. Values below the first resonance are sufficient but not necessary for invertibility.