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

Theorem: Self-adjoint microscopic Hamiltonian

One fixed interacting microscopic model › Construction of continuum dynamics · lines 499–549 · proof included or linked

Hawk02-E@1#thm:selfadjoint, Hawk02-E@1#eq:lower-H

The operator (eq:H) is self-adjoint on $H_{S} \otimes D (H_{B})$ , bounded below, and generates the unitary group $U (t) = e^{- i t H}$ . As quadratic forms,

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

Proof. The standard annihilation estimate, proved by Cauchy–Schwarz on each particle sector, is

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

The creation identity

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

therefore gives

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

For every $b > 0$ , the spectral theorem bounds $H_{B}^{1/2} ψ$ by $b ∥ H_{B} ψ ∥ + c_{b} ∥ ψ ∥$ . Thus $V$ has relative operator bound zero. The bounded source term does not change the domain, so Kato–Rellich establishes self-adjointness.

For a normalized vector in the form domain,

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

Since $H_{S} \geq 0$ , this proves (eq:lower-H). The unitary group follows from self-adjointness.

Displayed formulas in this passage

Hawk02-E@1#eq:lower-H · line 504
$H \geq \frac{1}{2} H_{B} - 2 G_{ω}^{2} I .$
unlabeled · line 513
$∥ a (f) ψ ∥ \leq h_{B}^{- 1/2} f H_{B}^{1/2} ψ .$
unlabeled · line 520
$a^{†} (f) ψ^{2} = ∥ a (f) ψ ∥^{2} + ∥ f ∥^{2} ∥ ψ ∥^{2}$
unlabeled · line 526
$∥ V ψ ∥ \leq 2 G_{ω} H_{B}^{1/2} ψ + G ∥ ψ ∥ .$
unlabeled · line 540
$∣ ⟨ V ⟩ ∣ \leq 2 G_{ω} ⟨ H_{B} ⟩ \leq \frac{1}{2} ⟨ H_{B} ⟩ + 2 G_{ω}^{2} .$

Theorem: Self-adjoint microscopic Hamiltonian

Displayed formulas in this passage

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