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

Theorem: Self-adjoint complete dynamics

An independently specified regulated microscopic model › Frozen source and finite-band reservoir · lines 1296–1342 · proof included or linked

Hawk04-B@2#thm:H, Hawk04-B@2#eq:formlower

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The Hamiltonian (eq:H) is self-adjoint on $H_{S} \otimes D (H_{B})$ , is bounded below, and satisfies the form inequality

H \geq \frac{1}{2} H_{B} - 2 G_{ω}^{2} I .

It generates $U (t) = e^{- i t H}$ . Every isometric diary loading with a fixed pure reservoir preparation is globally preserved at each finite time.

Proof. On the finite-particle core,

∥ a (f) ψ ∥ \leq h_{B}^{- 1/2} f H_{B}^{1/2} ψ,

and the CCR imply

a^{†} (f) ψ^{2} = ∥ a (f) ψ ∥^{2} + ∥ f ∥^{2} ∥ ψ ∥^{2} .

Consequently,

∥ V_{int} ψ ∥ \leq 2 G_{ω} H_{B}^{1/2} ψ + G ∥ ψ ∥ .

The square-root term has arbitrarily small relative bound with respect to $H_{B}$ . Since $H_{S}$ is bounded, the Kato–Rellich theorem gives self-adjointness on the stated domain Kato. These field estimates are also standard in spin–boson Hamiltonian constructions HaslerHerbst.

For normalized form-domain vectors,

∣ ⟨ V_{int} ⟩ ∣ \leq 2 G_{ω} ⟨ H_{B} ⟩ \leq \frac{1}{2} ⟨ H_{B} ⟩ + 2 G_{ω}^{2} .

Together with $H_{S} \geq 0$ , this proves (eq:formlower). The spectral theorem supplies the unitary group, and Theorem thm:global gives complete code preservation.

Displayed formulas in this passage

Hawk04-B@2#eq:formlower · line 1301
$H \geq \frac{1}{2} H_{B} - 2 G_{ω}^{2} I .$
unlabeled · line 1312
$∥ a (f) ψ ∥ \leq h_{B}^{- 1/2} f H_{B}^{1/2} ψ,$
unlabeled · line 1317
$a^{†} (f) ψ^{2} = ∥ a (f) ψ ∥^{2} + ∥ f ∥^{2} ∥ ψ ∥^{2} .$
unlabeled · line 1322
$∥ V_{int} ψ ∥ \leq 2 G_{ω} H_{B}^{1/2} ψ + G ∥ ψ ∥ .$
unlabeled · line 1334
$∣ ⟨ V_{int} ⟩ ∣ \leq 2 G_{ω} ⟨ H_{B} ⟩ \leq \frac{1}{2} ⟨ H_{B} ⟩ + 2 G_{ω}^{2} .$

Theorem: Self-adjoint complete dynamics

Displayed formulas in this passage

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