Manuscript IV@1 · IV.tex · lemma · 351cb2d7

Lemma: TT and finite-time gains

Model 2: spectral removal at fixed number cutoff › Transfer variables and the time factor · lines 1730–1769 · proof included or linked

IV@1#lem:gains, IV@1#eq:master2

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For every integer $n \geq 0$ ,

∣ J (k, ν) ∣ \leq C_{n} ⟨ k ⟩^{- n} (1 + ∣ ν ∣)^{- 1} .

For $k = ϵ p + δ q$ ,

Π (p) (q \otimes q) = Π (p) (k \otimes k), ∣Π (p) (k \otimes k) ∣^{2} = \frac{1}{2} (∣ k ∣^{2} - ∣ p \cdot k ∣^{2})^{2} .

Consequently, for every $n$ ,

∣ H^{ϵδ} (p, q) ∣ \leq \frac{C _{n} ⟨ ϵ p + δ q ⟩ ^{- n}}{∣ p ∣ E _{q} [ 1 + ∣ ϵ ∣ p ∣ + δ E _{q} ∣ ]} .

Proof. Spatial integrations by parts give arbitrary decay of $F$ and $\partial_{s} F$ , uniformly on the finite interval. Since $F$ vanishes near $ℓ$ , integration by parts in time for $∣ ν ∣ > 1$ gives

J (k, ν) = \frac{F ( 0 , k )}{i ν} + \frac{1}{i ν} \int_{0}^{ℓ} \partial_{s} F (s, k) e^{- i ν s} d s .

The initial endpoint has not been omitted. For $∣ ν ∣ \leq 1$ use the original integral.

The TT projection annihilates each term containing a factor $p$ . Substituting $δ q = k - ϵ p$ gives the identity. The norm is the squared norm of the trace-free part of $k_{⊥} \otimes k_{⊥}$ on a two-dimensional transverse plane. Absorb the factor $∣ k ∣^{2}$ into the arbitrary spatial decay.

Displayed formulas in this passage

IV@1#eq:master2 · line 1743
$∣ H^{ϵδ} (p, q) ∣ \leq \frac{C _{n} ⟨ ϵ p + δ q ⟩ ^{- n}}{∣ p ∣ E _{q} [ 1 + ∣ ϵ ∣ p ∣ + δ E _{q} ∣ ]} .$
unlabeled · line 1733
$∣ J (k, ν) ∣ \leq C_{n} ⟨ k ⟩^{- n} (1 + ∣ ν ∣)^{- 1} .$
unlabeled · line 1737
$Π (p) (q \otimes q) = Π (p) (k \otimes k), ∣Π (p) (k \otimes k) ∣^{2} = \frac{1}{2} (∣ k ∣^{2} - ∣ p \cdot k ∣^{2})^{2} .$
unlabeled · line 1757
$J (k, ν) = \frac{F ( 0 , k )}{i ν} + \frac{1}{i ν} \int_{0}^{ℓ} \partial_{s} F (s, k) e^{- i ν s} d s .$

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section: Transfer variables and the time factor