Manuscript Hawk02-E@1 · Hawk02-E.tex · section · 4841592b

Regulated scalar reservoir

One fixed interacting microscopic model › Regulated scalar reservoir · lines 404–479

Hawk02-E@1#eq:f-norm, Hawk02-E@1#eq:f-weighted, Hawk02-E@1#eq:H, Hawk02-E@1#eq:G, Hawk02-E@1#eq:Gomega

Fix

0 < μ < Λ < \infty, r \geq 1.

The reservoir one-particle space and Hamiltonian are

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

On bosonic Fock space,

H_{B} = F_{s} (h_{μ, Λ}), H_{B} = d Γ (h_{B}) .

The vacuum is denoted $Ω_{B}$ .

For species $a$ , define

f_{a} (ω) = \frac{e _{a}}{2 ω},

where $e_{a}$ is the species basis vector. Let

Φ (f) = a (f) + a^{†} (f),

with $a (f)$ conjugate-linear in $f$ . Then

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

The $1/ 2 ω$ factor specifies a regulated scalar-mode coupling. Its normalization is a model input, not an inferred bulk-to-boundary correspondence.

Freeze real $g_{1}, \dots, g_{r}$ , and define

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

Introduce the explicit constants

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

For the common operator (eq:coupling-A), these satisfy

G \leq a_{*} (a \sum ∣ g_{a} ∣) \frac{1}{2} lo g \frac{Λ}{μ},

and the corresponding bound for $G_{ω}$ follows from (eq:f-weighted).

Displayed formulas in this passage

Hawk02-E@1#eq:f-norm · line 439
$∥ f_{a} ∥^{2} h_{B}^{- 1/2} f_{a}^{2} = \frac{1}{2} lo g \frac{Λ}{μ}, = \frac{1}{2} (\frac{1}{μ} - \frac{1}{Λ}) .$
Hawk02-E@1#eq:f-weighted · line 439
$∥ f_{a} ∥^{2} h_{B}^{- 1/2} f_{a}^{2} = \frac{1}{2} lo g \frac{Λ}{μ}, = \frac{1}{2} (\frac{1}{μ} - \frac{1}{Λ}) .$
Hawk02-E@1#eq:H · line 453
$V = a = 1 \sum^{r} g_{a} A_{a} \otimes Φ (f_{a}), H = H_{S} \otimes I + I \otimes H_{B} + V .$
Hawk02-E@1#eq:G · line 460
$G G_{ω} = a = 1 \sum^{r} ∣ g_{a} ∣ ∥ A_{a} ∥ ∥ f_{a} ∥, = a = 1 \sum^{r} ∣ g_{a} ∣ ∥ A_{a} ∥ h_{B}^{- 1/2} f_{a} .$
Hawk02-E@1#eq:Gomega · line 460
$G G_{ω} = a = 1 \sum^{r} ∣ g_{a} ∣ ∥ A_{a} ∥ ∥ f_{a} ∥, = a = 1 \sum^{r} ∣ g_{a} ∣ ∥ A_{a} ∥ h_{B}^{- 1/2} f_{a} .$
unlabeled · line 407
$0 < μ < Λ < \infty, r \geq 1.$
unlabeled · line 413
$h_{μ, Λ} = L^{2} ([μ, Λ], d ω; C^{r}), h_{B} f (ω) = ω f (ω) .$
unlabeled · line 421
$H_{B} = F_{s} (h_{μ, Λ}), H_{B} = d Γ (h_{B}) .$
unlabeled · line 429
$f_{a} (ω) = \frac{e _{a}}{2 ω},$
unlabeled · line 434
$Φ (f) = a (f) + a^{†} (f),$
unlabeled · line 472
$G \leq a_{*} (a \sum ∣ g_{a} ∣) \frac{1}{2} lo g \frac{Λ}{μ},$