Manuscript Hawk02-D@1 · Hawk02-D.tex · section · e6e4b72f

Exact amplitudes and quantitative collection bounds

Exact amplitudes and quantitative collection bounds · lines 690–721

Hawk02-D@1#eq:local-amplitude-decay, Hawk02-D@1#eq:collection-tail, Hawk02-D@1#eq:complete-weight, Hawk02-D@1#eq:collected-weight, Hawk02-D@1#eq:eta

For each fixed $j$ , the spectral integral is the Fourier transform of an $L^{1}$ density. Therefore

u_{j} (t) ⟶ 0 (t ⟶ \infty) .

Define

ℓ_{M} (t) s_{M} (t) p_{M} (t) = j > M \sum ∣ u_{j} (t) ∣^{2}, = q (t) + ℓ_{M} (t), = j = 1 \sum^{M} ∣ u_{j} (t) ∣^{2} = 1 - s_{M} (t) .

Thus $s_{M}$ contains both source survival and every uncollected reservoir site.

For $x \geq 0$ , set

η_{M} (x) = min {1, e^{x} \frac{x ^{M + 1}}{( M + 1 )!}} .

Displayed formulas in this passage

Hawk02-D@1#eq:local-amplitude-decay · line 694
$u_{j} (t) ⟶ 0 (t ⟶ \infty) .$
Hawk02-D@1#eq:collection-tail · line 701
$ℓ_{M} (t) s_{M} (t) p_{M} (t) = j > M \sum ∣ u_{j} (t) ∣^{2}, = q (t) + ℓ_{M} (t), = j = 1 \sum^{M} ∣ u_{j} (t) ∣^{2} = 1 - s_{M} (t) .$
Hawk02-D@1#eq:complete-weight · line 701
$ℓ_{M} (t) s_{M} (t) p_{M} (t) = j > M \sum ∣ u_{j} (t) ∣^{2}, = q (t) + ℓ_{M} (t), = j = 1 \sum^{M} ∣ u_{j} (t) ∣^{2} = 1 - s_{M} (t) .$
Hawk02-D@1#eq:collected-weight · line 701
$ℓ_{M} (t) s_{M} (t) p_{M} (t) = j > M \sum ∣ u_{j} (t) ∣^{2}, = q (t) + ℓ_{M} (t), = j = 1 \sum^{M} ∣ u_{j} (t) ∣^{2} = 1 - s_{M} (t) .$
Hawk02-D@1#eq:eta · line 713
$η_{M} (x) = min {1, e^{x} \frac{x ^{M + 1}}{( M + 1 )!}} .$