Manuscript Hawk04-B@2 · Hawk04-B.tex · section · ba3b0713

Frozen source and finite-band reservoir

An independently specified regulated microscopic model › Frozen source and finite-band reservoir · lines 1222–1295

Hawk04-B@2#eq:H

Let $N = 2 n \geq 4$ , and let

χ_{i}^{†} = χ_{i}, {χ_{i}, χ_{j}} = 2 δ_{ij} I

act on $H_{S} = F_{f} (C^{n})$ . For frozen real coefficients $J_{ij k l}$ , set

H_{bare} = i < j < k < l \sum J_{ij k l} χ_{i} χ_{j} χ_{k} χ_{l}, B_{N} = i < j < k < l \sum ∣ J_{ij k l} ∣,

H_{S} = H_{bare} + B_{N} I, M_{N} = 2 B_{N} .

Each quartic monomial is self-adjoint with norm one, hence

0 \leq H_{S} \leq M_{N} I .

Define

S_{j} = i χ_{2 j - 1} χ_{2 j}, O_{s^{'}} = \frac{1}{n} j = 1 \sum^{n} s^{'}_{j} S_{j}, A = α O_{s^{'}} + ζ O_{s^{'}}^{2},

where $s^{'}_{j} \in {- 1, 1}$ and $α, ζ \in R$ . For the estimates, allow any fixed finite family of even self-adjoint source operators $A_{a}$ , $1 \leq a \leq r$ .

Fix

0 < μ < Λ < \infty, h = L^{2} ([μ, Λ], d ω; C^{r}), (h_{B} f) (ω) = ω f (ω) .

On $F_{s} (h)$ , let

H_{B} = d Γ (h_{B}), f_{a} (ω) = \frac{e _{a}}{2 ω}, Φ (f) = a (f) + a^{†} (f) .

Annihilation and the first slot of the one-particle inner product are conjugate-linear.

For frozen real $g_{a}$ , set

H = H_{S} + H_{B} + V_{int}, V_{int} = a = 1 \sum^{r} g_{a} A_{a} \otimes Φ (f_{a}) .

Identity factors are suppressed. Define

G = a \sum ∣ g_{a} ∣ ∥ A_{a} ∥ ∥ f_{a} ∥, G_{ω} = a \sum ∣ g_{a} ∣ ∥ A_{a} ∥ h_{B}^{- 1/2} f_{a} .

The form-factor norms are

∥ f_{a} ∥^{2} = \frac{1}{2} lo g \frac{Λ}{μ}, h_{B}^{- 1/2} f_{a}^{2} = \frac{1}{2} (\frac{1}{μ} - \frac{1}{Λ}) .

Displayed formulas in this passage

Hawk04-B@2#eq:H · line 1275
$H = H_{S} + H_{B} + V_{int}, V_{int} = a = 1 \sum^{r} g_{a} A_{a} \otimes Φ (f_{a}) .$
unlabeled · line 1225
$χ_{i}^{†} = χ_{i}, {χ_{i}, χ_{j}} = 2 δ_{ij} I$
unlabeled · line 1231
$H_{bare} = i < j < k < l \sum J_{ij k l} χ_{i} χ_{j} χ_{k} χ_{l}, B_{N} = i < j < k < l \sum ∣ J_{ij k l} ∣,$
unlabeled · line 1237
$H_{S} = H_{bare} + B_{N} I, M_{N} = 2 B_{N} .$
unlabeled · line 1242
$0 \leq H_{S} \leq M_{N} I .$
unlabeled · line 1247
$S_{j} = i χ_{2 j - 1} χ_{2 j}, O_{s^{'}} = \frac{1}{n} j = 1 \sum^{n} s^{'}_{j} S_{j}, A = α O_{s^{'}} + ζ O_{s^{'}}^{2},$
unlabeled · line 1258
$0 < μ < Λ < \infty, h = L^{2} ([μ, Λ], d ω; C^{r}), (h_{B} f) (ω) = ω f (ω) .$
unlabeled · line 1265
$H_{B} = d Γ (h_{B}), f_{a} (ω) = \frac{e _{a}}{2 ω}, Φ (f) = a (f) + a^{†} (f) .$
unlabeled · line 1282
$G = a \sum ∣ g_{a} ∣ ∥ A_{a} ∥ ∥ f_{a} ∥, G_{ω} = a \sum ∣ g_{a} ∣ ∥ A_{a} ∥ h_{B}^{- 1/2} f_{a} .$
unlabeled · line 1289
$∥ f_{a} ∥^{2} = \frac{1}{2} lo g \frac{Λ}{μ}, h_{B}^{- 1/2} f_{a}^{2} = \frac{1}{2} (\frac{1}{μ} - \frac{1}{Λ}) .$

Frozen source and finite-band reservoir

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