= Discrete Helmholtz resonance
{title2=$\lambda_{pq}=4h^{-2}[\sin^2(p\pi h/2)+\sin^2(q\pi h/2)]$}
On a square Dirichlet grid with spacing $h=1/J$ and interior indices $1,\ldots,J-1$, 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 $1\leq p,q\leq J-1$. Values below the first resonance are sufficient but not necessary for invertibility.
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