Manuscript Em03-G@1 · Em03-G-v1.tex · section · 0bb3a8a3

Explicit local sine-probe parameters

Explicit local sine-probe parameters · lines 2310–2410

This appendix specifies the local quantities used in Theorem thm:local. It specializes the compact linear system–probe construction of Em03-D and proves the sine-effect adaptation.

Fix a compact witness test $B = F [b]$ constructed after the source slabs. Choose a spacelike-compact free probe solution $v_{bulk}$ equal to one near $supp B$ . Compact Cauchy data equal to the constant solution on the causal predecessor of that set give it by finite propagation. Write $v_{bulk} = Y_{0} v / r$ .

Choose a switch $η_{+}$ in a compact future slab, zero before the slab and one after it. Define

z_{red} = - D (η_{+} v), z = F [z_{red}] .

Then $G_{A} z = (1 - η_{+}) v_{bulk} = 1$ on the coupling support.

Use

S_{probe, int} = - ϵ \int ρϕ ψ d vol_{g}, ρ = - B,

T_{ϵ} = (P ϵ ρ ϵ ρ P) .

Set

w_{ϵ} = G_{A, T_{ϵ}} (0, z)^{T}, B_{ϵ} = - ρ (w_{ϵ})_{ψ}, z_{ϵ} = z / ϵ - ρ (w_{ϵ})_{ϕ} .

Advanced scattering gives

Θ_{ϵ} (I \otimes W_{ψ} (z / ϵ)) = W_{ϕ} (B_{ϵ}) \otimes W_{ψ} (z_{ϵ}) .

It follows by applying

θ_{ϵ} H = H - (T_{ϵ} - P \oplus P) G_{A, T_{ϵ}} H

to $H = (0, z / ϵ)$ . Green's identity and conserved coupled symplectic current verify this test map and its Weyl morphism.

For any centered quasifree independent probe,

ε_{σ, ϵ} (O_{s i n, ϵ}) = \frac{1}{2} [I + c_{ϵ} Im W_{ϕ} (B_{ϵ})] .

The probe characteristic coefficient is real, so taking the imaginary part commutes with the probe expectation. No mixed tensor-factor Weyl phase is present.

Here are finite quantitative constants. If the coupling is supported in $I \times [- X, X]$ , let $Δ = ∣ I ∣$ , $V_{K} = 2 X Δ$ , and

a_{p} = f ρ = - Y_{0} b / r, A_{p} u = G_{A, D} (a_{p} u), v_{A} = G_{A, D} z_{red} .

On the coupling set, $- a_{p} v_{A} = b$ .

Use

∥ u ∥_{X}^{2} = t \in I sup j = 0 \sum^{2} (∥ \partial_{t} \partial_{x}^{j} u ∥_{2}^{2} + ∥ \partial_{x}^{j + 1} u ∥_{2}^{2} + ∥ \partial_{x}^{j} u ∥_{2}^{2}) .

Choose

c_{2} = 64 (1 + j = 0 \sum^{2} ∥ \partial_{x}^{j} V ∥_{\infty}),

M_{2} = 8 t \in I sup j = 0 \sum^{2} ∥ \partial_{x}^{j} a_{p} ∥_{\infty}, Λ = Δ e^{c_{2} Δ} M_{2} .

Spatial wave-energy differentiation through order two, the product rule, and backward Gronwall inequality give

∥ A_{p} u ∥_{X} \leq Λ∥ u ∥_{X} .

Displayed formulas in this passage

unlabeled · line 2327
$z_{red} = - D (η_{+} v), z = F [z_{red}] .$
unlabeled · line 2335
$S_{probe, int} = - ϵ \int ρϕ ψ d vol_{g}, ρ = - B,$
unlabeled · line 2340
$T_{ϵ} = (P ϵ ρ ϵ ρ P) .$
unlabeled · line 2346
$w_{ϵ} = G_{A, T_{ϵ}} (0, z)^{T}, B_{ϵ} = - ρ (w_{ϵ})_{ψ}, z_{ϵ} = z / ϵ - ρ (w_{ϵ})_{ϕ} .$
unlabeled · line 2354
$Θ_{ϵ} (I \otimes W_{ψ} (z / ϵ)) = W_{ϕ} (B_{ϵ}) \otimes W_{ψ} (z_{ϵ}) .$
unlabeled · line 2359
$θ_{ϵ} H = H - (T_{ϵ} - P \oplus P) G_{A, T_{ϵ}} H$
unlabeled · line 2367
$ε_{σ, ϵ} (O_{s i n, ϵ}) = \frac{1}{2} [I + c_{ϵ} Im W_{ϕ} (B_{ϵ})] .$
unlabeled · line 2378
$a_{p} = f ρ = - Y_{0} b / r, A_{p} u = G_{A, D} (a_{p} u), v_{A} = G_{A, D} z_{red} .$
unlabeled · line 2387
$∥ u ∥_{X}^{2} = t \in I sup j = 0 \sum^{2} (∥ \partial_{t} \partial_{x}^{j} u ∥_{2}^{2} + ∥ \partial_{x}^{j + 1} u ∥_{2}^{2} + ∥ \partial_{x}^{j} u ∥_{2}^{2}) .$
unlabeled · line 2397
$c_{2} = 64 (1 + j = 0 \sum^{2} ∥ \partial_{x}^{j} V ∥_{\infty}),$
unlabeled · line 2400
$M_{2} = 8 t \in I sup j = 0 \sum^{2} ∥ \partial_{x}^{j} a_{p} ∥_{\infty}, Λ = Δ e^{c_{2} Δ} M_{2} .$
unlabeled · line 2408
$∥ A_{p} u ∥_{X} \leq Λ∥ u ∥_{X} .$

Explicit local sine-probe parameters

Displayed formulas in this passage

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