Modified Helmholtz adjoint plane wave ($v_{\lambda }$)

For $k>0$, a nonzero spectral parameter for a linear boundary value problem givesThe identity $A^2+B^2=k^2$ proves that this solves the modified Helmholtz equation. The reciprocal companion is $v_{1/\lambda }=\overline{v_{\overline{\lambda }}}$.

On a square, a normalized modified Helmholtz adjoint plane wave with positive normal growth toward one side has magnitude one there, magnitude $e^{-2\eta }$ on the opposite side, and an adjacent-side magnitude integral bounded by $1/\eta$. Thus large tangential spectral frequencies suppress remote-side normal-trace coefficients. Paired sine collocation of square modified Helmholtz global relations further cancels adjacent-side unknown traces exactly.