Manuscript Em03-A@1 · Em03-A-v1.tex · proposition · 6d54627d

Proposition: Mode-window number rate

Outgoing observables, rate normalization, and error bounds › A precise 1/(2π) counting convention · lines 1032–1068 · proof included or linked

Em03-A@1#prop:rate, Em03-A@1#eq:windowcount

The operator $K_{B, T}$ is positive and trace class, and

N_{B, T} = \frac{T}{2 π} \int_{B} \frac{Γ _{ℓ} ( ω )}{e ^{β_{H} ω} - 1} d ω .

Thus the rate density under Definition def:window is (eq:rate).

Proof. The integral kernel of $A_{B, T}$ is

(2 π)^{- 1/2} e^{- i ω t} 1_{B} (ω) .

It is Hilbert–Schmidt, with

∥ A_{B, T} ∥_{HS}^{2} = \frac{T ∣ B ∣}{2 π} .

Consequently $K_{B, T} = A_{B, T}^{*} A_{B, T}$ is positive and trace class.

The function $n_{out}$ is nonnegative and bounded on $B$ . Therefore

Tr (M_{n_{out}} K_{B, T}) = A_{B, T} M_{n_{out}}_{HS}^{2} = \frac{T}{2 π} \int_{B} n_{out} (ω) d ω .

Substitution of its proved value gives (eq:windowcount). Dividing by $T$ identifies the density.

Displayed formulas in this passage

Em03-A@1#eq:windowcount · line 1035
$N_{B, T} = \frac{T}{2 π} \int_{B} \frac{Γ _{ℓ} ( ω )}{e ^{β_{H} ω} - 1} d ω .$
unlabeled · line 1048
$(2 π)^{- 1/2} e^{- i ω t} 1_{B} (ω) .$
unlabeled · line 1052
$∥ A_{B, T} ∥_{HS}^{2} = \frac{T ∣ B ∣}{2 π} .$
unlabeled · line 1059
$Tr (M_{n_{out}} K_{B, T}) = A_{B, T} M_{n_{out}}_{HS}^{2} = \frac{T}{2 π} \int_{B} n_{out} (ω) d ω .$