Manuscript Hawk04-D@1 · Hawk04-D-v1.tex · lemma · 39b16f67

Lemma: Complete dynamics exists

A model-specific finite-band regulator · lines 1018–1055 · proof included or linked

Hawk04-D@1#lem:H

The Hamiltonian is self-adjoint on the free bath domain and generates a complete unitary group. If $H_{S} \geq 0$ , it has the form lower bound

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

Proof. The field estimates give

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

and

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

Hence the interaction obeys

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

The square-root term has arbitrarily small relative operator bound with respect to $H_{B}$ . Kato–Rellich therefore gives self-adjointness Kato; these are standard field-operator estimates HaslerHerbst. For normalized form-domain vectors,

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

The spectral theorem supplies the unitary group.

Displayed formulas in this passage

unlabeled · line 1023
$H \geq \frac{1}{2} H_{B} - 2 G_{ω}^{2} I .$
unlabeled · line 1030
$∥ a (f) ψ ∥ \leq h^{- 1/2} f H_{B}^{1/2} ψ,$
unlabeled · line 1035
$a^{†} (f) ψ^{2} = ∥ a (f) ψ ∥^{2} + ∥ f ∥^{2} ∥ ψ ∥^{2} .$
unlabeled · line 1040
$∥ V_{int} ψ ∥ \leq 2 G_{ω} H_{B}^{1/2} ψ + G ∥ ψ ∥ .$
unlabeled · line 1049
$∣ ⟨ V_{int} ⟩ ∣ \leq 2 G_{ω} ⟨ H_{B} ⟩ \leq \frac{1}{2} ⟨ H_{B} ⟩ + 2 G_{ω}^{2} .$

Lemma: Complete dynamics exists

Displayed formulas in this passage

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