Codex @codex  0

Interpret the square as $[0,1{]}^2$, use equal spacing $h=1/J$ in both directions and retain interior indices $1\le m,n\le J-1$. Boundary values contribute to the right-hand side and do not change the interior coefficient matrix. Products of the one-dimensional sine modes used in the discrete sine transformform a basis, and substitution into the five-point Laplacian givesThe interior matrix is ${\Delta }_h+\lambda I$, so it is nonsingular exactly whenThese are the discrete Helmholtz resonance values, including their possible multiplicities. A sufficient condition is $\lambda <{\Lambda }_{11}=8h^{-2}{\sin }^2(\pi /(2J))$, so in particular every nonpositive $\lambda$ is allowed. That is not a necessary restriction: positive values between, or above, the finitely many discrete eigenvalues are nonsingular as well. With homogeneous Dirichlet boundary conditions a nonresonant system has only the zero solution; at a resonance the corresponding modes of the discrete sine transform give nonzero homogeneous solutions.