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

Explicit local sine-probe parameters

Explicit local sine-probe parameters · lines 2410–2506

For a test supported in $∣ x ∣ \leq X$ , the compact-mode estimate in Lemma lem:compactregularity gives

∥ k (f) ∥_{U} \leq B_{X} (f),

B_{X} (f)^{2} = \frac{C _{X} ^{2}}{2 π} (A_{0} ∥ f ∥_{1}^{2} + A_{1} ∥ \partial_{t} f ∥_{1}^{2}),

A_{0} = \frac{1}{2} + \frac{2}{β ω _{c}}, A_{1} = \frac{1}{2 ω _{c} ^{2}} + \frac{2}{3 β ω _{c} ^{3}} .

The constants are obtained by integrating the low-frequency $O (ω)$ and high-frequency $O (ω^{- 3/2})$ coefficient estimates, retaining the horizon weight.

With

a_{0} = ∥ a_{p} ∥_{\infty}, a_{t} = ∥ \partial_{t} a_{p} ∥_{\infty},

put

L_{p} = \frac{2 V _{K} C _{X}}{2 π} A_{0} a_{0}^{2} + A_{1} (a_{t} + a_{0})^{2} .

The one-dimensional Sobolev inequality gives

∥ k (a_{p} u) ∥_{U} \leq L_{p} ∥ u ∥_{X} .

When $ϵ^{2} Λ^{2} < 1$ , the exact reduced solution is

(w_{ϵ})_{ψ} = (I - ϵ^{2} A_{p}^{2})^{- 1} v_{A}, (w_{ϵ})_{ϕ} = - ϵ A_{p} (w_{ϵ})_{ψ} .

The Neumann series converges in $X$ , has zero final data, and solves the coupled equation; Cauchy uniqueness identifies it with the exact advanced solution. Consequently

d_{trf} (L, ϵ) = L_{p} \frac{ϵ ^{2} Λ ^{2}}{1 - ϵ ^{2} Λ ^{2}} ∥ v_{A} ∥_{X}

bounds $∥ k (B_{ϵ}) - l_{L} ∥_{U}$ .

The residual probe test is

(z_{ϵ})_{red} = z_{red} / ϵ + ϵ j_{ϵ}, j_{ϵ} = a_{p} A_{p} (I - ϵ^{2} A_{p}^{2})^{- 1} v_{A} .

For $ϵ^{2} Λ^{2} \leq 1/2$ ,

∥ k (j_{ϵ}) ∥_{U} \leq b_{J} := 2 L_{p} Λ∥ v_{A} ∥_{X} .

Take an independent reference probe with Unruh covariance. Choose a real compact test $p_{0}$ with $c_{0} = s_{ψ} (z, p_{0}) \neq = 0$ . For small enough $ϵ$ , set

p_{ϵ} = p_{0} / s_{ψ} (z_{ϵ}, p_{0}) .

Define the symplectic transformation

S_{ϵ} u = u - [s_{ψ} (u, p_{ϵ}) z_{ϵ} - s_{ψ} (u, z_{ϵ}) p_{ϵ}] + ϵ^{2} s_{ψ} (u, p_{ϵ}) z_{ϵ} - ϵ^{- 2} s_{ψ} (u, z_{ϵ}) p_{ϵ} .

It squeezes $z_{ϵ}$ by $ϵ^{2}$ , squeezes its conjugate by $ϵ^{- 2}$ , and fixes the symplectic orthogonal complement. Prepare

σ_{L, ϵ} (W_{ψ} (u)) = σ_{ref} (W_{ψ} (S_{ϵ} u)) .

Symplecticity proves positivity and quasifreeness. The finite-rank covariance change is smooth, by compact-test smearing of the reference Hadamard bisolution; its Hadamard singularity is unchanged.

Displayed formulas in this passage

unlabeled · line 2414
$∥ k (f) ∥_{U} \leq B_{X} (f),$
unlabeled · line 2417
$B_{X} (f)^{2} = \frac{C _{X} ^{2}}{2 π} (A_{0} ∥ f ∥_{1}^{2} + A_{1} ∥ \partial_{t} f ∥_{1}^{2}),$
unlabeled · line 2422
$A_{0} = \frac{1}{2} + \frac{2}{β ω _{c}}, A_{1} = \frac{1}{2 ω _{c} ^{2}} + \frac{2}{3 β ω _{c} ^{3}} .$
unlabeled · line 2431
$a_{0} = ∥ a_{p} ∥_{\infty}, a_{t} = ∥ \partial_{t} a_{p} ∥_{\infty},$
unlabeled · line 2436
$L_{p} = \frac{2 V _{K} C _{X}}{2 π} A_{0} a_{0}^{2} + A_{1} (a_{t} + a_{0})^{2} .$
unlabeled · line 2442
$∥ k (a_{p} u) ∥_{U} \leq L_{p} ∥ u ∥_{X} .$
unlabeled · line 2448
$(w_{ϵ})_{ψ} = (I - ϵ^{2} A_{p}^{2})^{- 1} v_{A}, (w_{ϵ})_{ϕ} = - ϵ A_{p} (w_{ϵ})_{ψ} .$
unlabeled · line 2456
$d_{trf} (L, ϵ) = L_{p} \frac{ϵ ^{2} Λ ^{2}}{1 - ϵ ^{2} Λ ^{2}} ∥ v_{A} ∥_{X}$
unlabeled · line 2465
$(z_{ϵ})_{red} = z_{red} / ϵ + ϵ j_{ϵ}, j_{ϵ} = a_{p} A_{p} (I - ϵ^{2} A_{p}^{2})^{- 1} v_{A} .$
unlabeled · line 2474
$∥ k (j_{ϵ}) ∥_{U} \leq b_{J} := 2 L_{p} Λ∥ v_{A} ∥_{X} .$
unlabeled · line 2483
$p_{ϵ} = p_{0} / s_{ψ} (z_{ϵ}, p_{0}) .$
unlabeled · line 2487
$S_{ϵ} u = u - [s_{ψ} (u, p_{ϵ}) z_{ϵ} - s_{ψ} (u, z_{ϵ}) p_{ϵ}] + ϵ^{2} s_{ψ} (u, p_{ϵ}) z_{ϵ} - ϵ^{- 2} s_{ψ} (u, z_{ϵ}) p_{ϵ} .$
unlabeled · line 2499
$σ_{L, ϵ} (W_{ψ} (u)) = σ_{ref} (W_{ψ} (S_{ϵ} u)) .$

Explicit local sine-probe parameters

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