Manuscript En02-E@2 · En02-E-v2.tex · section · c2769986

The complete finite packet state

The complete finite packet state · lines 358–385

En02-E@2#eq:K

Fix $B = [ω_{-}, ω_{+}] \subset (0, \infty)$ and an orthonormal family $h_{1}, \dots, h_{J} \in L^{2} (B, d ω)$ . Define

b_{i} = a_{out} (h_{i}) = \int_{B} h_{i} (ω) a_{out} (ω) d ω .

Bounded covariance on $B$ extends the smooth-test formulas to these $L^{2}$ packets.

We distinguish

C_{ij} K_{ij} = ⟨ b_{i}^{†} b_{j} ⟩ = \int_{B} h_{i} h_{j} q_{out} d ω, = C_{j i} = \int_{B} h_{i} h_{j} q_{out} d ω .

Thus $K = C^{T}$ . We use $K$ as the ordinary one-particle operator matrix. Put

μ = tr K, q_{m a x} = B sup q_{out} < \infty; 0 ⪯ K ⪯ q_{m a x} I_{J} .

Displayed formulas in this passage

En02-E@2#eq:K · line 371
$C_{ij} K_{ij} = ⟨ b_{i}^{†} b_{j} ⟩ = \int_{B} h_{i} h_{j} q_{out} d ω, = C_{j i} = \int_{B} h_{i} h_{j} q_{out} d ω .$
unlabeled · line 363
$b_{i} = a_{out} (h_{i}) = \int_{B} h_{i} (ω) a_{out} (ω) d ω .$
unlabeled · line 381
$μ = tr K, q_{m a x} = B sup q_{out} < \infty; 0 ⪯ K ⪯ q_{m a x} I_{J} .$