Proposition: Variation with on-shell zero, edge, and switch separated

For a compact parameter $\xi$ in the slab,

where

The first term lies in ${\mathfrak{J}}_0$. The second is the coordinate edge term. The last displays the factors $1-g$ and $\partial g$.

Proof. The changed test is

Write ${\mathsf{L}}_{1,g}[K\xi ]={\mathsf{Q}}_{K\xi }+{\Sigma }_{g,\xi }$. Using $P_m{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f=f$, Lemma lem:Q gives

Subtract this from

The resulting identity is (eq:variation). The function $X_{\xi }{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f$ is smooth and compactly supported, so its $P_m$-test belongs to the equation ideal.