Manuscript XVI-D@1 · XVI-D.tex · proposition · b73c6f4a

Proposition: Scalar completion is verified on the witness

Completion formulas and the detailed domain check › Scalar forcing · lines 1081–1132 · proof included or linked

XVI-D@1#lem:scalar, XVI-D@1#eq:phicand, XVI-D@1#eq:phi-equation

The forcing

V_{1, g} [R_{λ}] ϕ

has real and imaginary parts in $D_{int}$ and support contained in $supp g$ . Hence

ϕ_{cand} := - E_{m}^{R} (V_{1, g} [R_{λ}] ϕ)

is defined by complex linear extension of the compact-source operator. It is smooth, has zero initial data, and satisfies

(□ + m^{2}) ϕ_{cand} = - V_{1, g} [R_{λ}] ϕ .

Proof. Expanding the divergence gives

V_{1, g} [R_{λ}] ϕ = (\partial_{μ} g) a_{1}^{μν} [R_{λ}] \partial_{ν} ϕ + g (\partial_{μ} a_{1}^{μν} [R_{λ}]) \partial_{ν} ϕ + g a_{1}^{μν} [R_{λ}] \partial_{μ} \partial_{ν} ϕ + m^{2} g b_{1} [R_{λ}] ϕ .

Every coefficient apart from $g$ and its derivatives is smooth. The support of every derivative of $g$ is contained in $supp g$ , a compact subset of $M_{T}^{\circ}$ . Thus every term is smooth and compactly supported there.

For real $F_{1}, F_{2} \in D_{int}$ , define

E_{m}^{R} (F_{1} + i F_{2}) = E_{m}^{R} F_{1} + i E_{m}^{R} F_{2} .

This is the unique complex linear extension. The real differential operator $□ + m^{2}$ , the initial data, and the support bounds pass to this extension. Equation (eq:phi-equation) includes the minus sign in (eq:phicand).

Displayed formulas in this passage

XVI-D@1#eq:phicand · line 1089
$ϕ_{cand} := - E_{m}^{R} (V_{1, g} [R_{λ}] ϕ)$
XVI-D@1#eq:phi-equation · line 1096
$(□ + m^{2}) ϕ_{cand} = - V_{1, g} [R_{λ}] ϕ .$
unlabeled · line 1084
$V_{1, g} [R_{λ}] ϕ$
unlabeled · line 1105
$V_{1, g} [R_{λ}] ϕ = (\partial_{μ} g) a_{1}^{μν} [R_{λ}] \partial_{ν} ϕ + g (\partial_{μ} a_{1}^{μν} [R_{λ}]) \partial_{ν} ϕ + g a_{1}^{μν} [R_{λ}] \partial_{μ} \partial_{ν} ϕ + m^{2} g b_{1} [R_{λ}] ϕ .$
unlabeled · line 1123
$E_{m}^{R} (F_{1} + i F_{2}) = E_{m}^{R} F_{1} + i E_{m}^{R} F_{2} .$

Nearby in this section

section: Scalar forcing