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

Outgoing packet coordinates

The complete scalar radiation channel › Outgoing packet coordinates · lines 721–754

Stationary scattering gives

b_{h} = a_{out} (h) = a (v_{h}), v_{h} = (τ h, ρ_{I} h) .

For normalized $h$ ,

∥ v_{h} ∥_{0} = 1, ∥ v_{h} ∥_{U}^{2} = 1 + 2 ν_{h},

ν_{h} = \int ∣ h (ω) ∣^{2} \frac{∣ τ ( ω ) ∣ ^{2}}{e ^{β ω} - 1} d ω .

The complete outgoing coefficient of pulse $j$ is

d_{j} = S a_{j},

where $S (ω)$ is the unitary scattering matrix. Its infinity component is

d_{j, \infty} = τ a_{j, H} + ρ_{I} a_{j, I} .

Thus

⟨ v_{h}, a_{j} ⟩ = \int h d_{j, \infty} d ω .

Every selected mode below is an outgoing mode of this scalar, not an additional radiation payload.

Displayed formulas in this passage

unlabeled · line 724
$b_{h} = a_{out} (h) = a (v_{h}), v_{h} = (τ h, ρ_{I} h) .$
unlabeled · line 729
$∥ v_{h} ∥_{0} = 1, ∥ v_{h} ∥_{U}^{2} = 1 + 2 ν_{h},$
unlabeled · line 733
$ν_{h} = \int ∣ h (ω) ∣^{2} \frac{∣ τ ( ω ) ∣ ^{2}}{e ^{β ω} - 1} d ω .$
unlabeled · line 738
$d_{j} = S a_{j},$
unlabeled · line 743
$d_{j, \infty} = τ a_{j, H} + ρ_{I} a_{j, I} .$
unlabeled · line 748
$⟨ v_{h}, a_{j} ⟩ = \int h d_{j, \infty} d ω .$