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

Definition: Time–frequency number window

Outgoing observables, rate normalization, and error bounds › A precise 1/(2π) counting convention · lines 1005–1030

Em03-A@1#def:window

On the one-particle space $L^{2} ((0, \infty), d ω)$ , let $P_{B}$ denote multiplication by the indicator of a compact positive-frequency interval $B$ . For $T > 0$ , define

(A_{B, T} h) (t) = \frac{1}{2 π} \int_{B} e^{- i ω t} h (ω) d ω, - T /2 \leq t \leq T /2,

and

K_{B, T} = A_{B, T}^{*} A_{B, T} .

The expected number in this mode window is

N_{B, T} := Tr (M_{n_{out}} K_{B, T}), n_{out} (ω) = Γ_{ℓ} (ω) n_{H} (ω),

where $M_{n_{out}}$ is the multiplication operator. Equivalently, this is the monotone expectation of finite-rank positive number-functionals approximating $K_{B, T}$ .

Displayed formulas in this passage

unlabeled · line 1010
$(A_{B, T} h) (t) = \frac{1}{2 π} \int_{B} e^{- i ω t} h (ω) d ω, - T /2 \leq t \leq T /2,$
unlabeled · line 1017
$K_{B, T} = A_{B, T}^{*} A_{B, T} .$
unlabeled · line 1021
$N_{B, T} := Tr (M_{n_{out}} K_{B, T}), n_{out} (ω) = Γ_{ℓ} (ω) n_{H} (ω),$

Definition: Time–frequency number window

Displayed formulas in this passage

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