Manuscript VII-H@1 · VII-H.tex · lemma · d20d2a9f

Lemma: Annular cone pairing

A delayed source/readout realization › A useful cone estimate · lines 1131–1170 · proof included or linked

VII-H@1#lem:annular, VII-H@1#eq:annular-pairing, VII-H@1#eq:annular-bound

Let $G$ be a smooth compact test supported at positive times away from the origin. On the cone,

⟨ R^{\circ}, G ⟩ = \frac{ℏ}{16 π ^{3}} \int_{0}^{\infty} \int_{S^{2}} [\frac{G ( t , t n )}{t} + \partial_{r} G (t, t n)] d n d t .

If its cone sampling lies in $t \in [d, D]$ , $d > 0$ , then

∣ ⟨ R^{\circ}, G ⟩ ∣ \leq \frac{ℏ ( D - d )}{4 π ^{2}} [d^{- 1} ∥ G ∥_{\infty} + ∥ \partial_{r} G ∥_{\infty}] .

Proof. For each $t, n$ , the substitution $u = t^{2} - r^{2}$ changes $r^{2} d r$ into $r ∣ d u ∣/2$ . The negative derivative at zero of $r G (t, r n) /2$ is

\frac{G ( t , t n )}{4 t} + \frac{\partial _{r} G ( t , t n )}{4} .

This proves (eq:annular-pairing). Use $area (S^{2}) = 4 π$ to obtain the bound.

Displayed formulas in this passage

VII-H@1#eq:annular-pairing · line 1135
$⟨ R^{\circ}, G ⟩ = \frac{ℏ}{16 π ^{3}} \int_{0}^{\infty} \int_{S^{2}} [\frac{G ( t , t n )}{t} + \partial_{r} G (t, t n)] d n d t .$
VII-H@1#eq:annular-bound · line 1147
$∣ ⟨ R^{\circ}, G ⟩ ∣ \leq \frac{ℏ ( D - d )}{4 π ^{2}} [d^{- 1} ∥ G ∥_{\infty} + ∥ \partial_{r} G ∥_{\infty}] .$
unlabeled · line 1164
$\frac{G ( t , t n )}{4 t} + \frac{\partial _{r} G ( t , t n )}{4} .$