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

Lemma: Compact coefficient approximation

Compact ordered realization and source-input stability › Filtered compact realization of a band coefficient · lines 1340–1388 · proof included or linked

Em03-G@1#lem:bridge

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The real compact test just defined satisfies

∥ k (f_{L, T}^{u, t_{j}}) - u ∥_{U} \leq Δ_{L} (u, t_{j}), Δ_{L} (u, t_{j}) ⟶ 0,

where, with $δ q = q_{L} - q, δ p = p_{L} - p$ ,

Δ_{L}^{2} = ∥ L^{3/4} δ p ∥_{2}^{2} + ∥ L^{5/4} δ q ∥_{2}^{2} + \frac{2}{β} (∥ L^{1/2} δ p ∥_{2}^{2} + ∥ L δ q ∥_{2}^{2}) .

Proof. The time-slice identity is

ϕ (F [D (η_{T} s_{L})]) = \int (p_{L} Ψ (t_{j}) - q_{L} \partial_{t} Ψ (t_{j})) d x .

Integration by parts proves it using the conserved Cauchy pairing and $\int η_{T}^{'} = 1$ . Applying $- \partial_{t}^{2}$ multiplies its coefficient by $ω^{2}$ . Therefore the coefficient error is

e^{i ω t_{j}} ω^{2} \frac{U δ p - i ω U δ q}{2 ω} .

Use $∣ z - i ω w ∣^{2} \leq 2∣ z ∣^{2} + 2 ω^{2} ∣ w ∣^{2}$ and $1 + 2 n_{H} \leq 1 + 2/ (β ω)$ . Spectral unitarity gives the displayed bound.

The cutoffs converge in $H^{2}$ for $p$ and $H^{3}$ for $q$ . Derivative tails and $\partial_{x}^{r} χ (x / L) = O (L^{- r})$ prove this. The potential and its derivatives are bounded. The fractional norms are controlled by these integer graph norms, using $λ^{3/2}, λ \leq 1 + λ^{2}$ and $λ^{5/2} \leq 1 + λ^{3}$ . Hence $Δ_{L} \to 0$ .

Displayed formulas in this passage

unlabeled · line 1343
$∥ k (f_{L, T}^{u, t_{j}}) - u ∥_{U} \leq Δ_{L} (u, t_{j}), Δ_{L} (u, t_{j}) ⟶ 0,$
unlabeled · line 1348
$Δ_{L}^{2} = ∥ L^{3/4} δ p ∥_{2}^{2} + ∥ L^{5/4} δ q ∥_{2}^{2} + \frac{2}{β} (∥ L^{1/2} δ p ∥_{2}^{2} + ∥ L δ q ∥_{2}^{2}) .$
unlabeled · line 1361
$ϕ (F [D (η_{T} s_{L})]) = \int (p_{L} Ψ (t_{j}) - q_{L} \partial_{t} Ψ (t_{j})) d x .$
unlabeled · line 1369
$e^{i ω t_{j}} ω^{2} \frac{U δ p - i ω U δ q}{2 ω} .$

Lemma: Compact coefficient approximation

Displayed formulas in this passage

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