Manuscript Em05-A@1 · Em05-A-v1.tex · section · 9cac0def

Continuum interpretation and endpoint attainability

Continuum interpretation and endpoint attainability · lines 1310–1355

The finite family includes $η = 1$ as an exact mathematical endpoint. It does not imply that the retained-storage continuum experiment of Em03-H attains that endpoint.

For clarity, the obstruction can be restated independently. Let

h = L^{2} (R_{+}), (h ψ) (ω) = ω ψ (ω), f (ω) = 2 e^{- ω} .

Take $t_{0} \geq 0$ , a initially vacant bath level, and the initial one-species state

ψ (0, ω) = e^{- i ω t_{0}} f (ω), B (0) = 0.

For finite extraction time $T$ , use bounded piecewise continuous real controls $v (u), κ (u)$ and the generator

K (u) = (h + v (u) ∣ f ⟩ ⟨ f ∣ κ (u) ⟨ f ∣ κ (u) ∣ f ⟩ 1)

on $h \oplus C$ .

Relative to $h \oplus 1$ , the perturbation is bounded, self-adjoint, and finite rank. Its interaction-picture Dyson series gives a unitary propagator. Put

A (u) = ⟨ f, ψ (u)⟩, c (u) = v (u) A (u) + κ (u) B (u) .

Variation of constants gives

ψ (T, ω) = f (ω) e^{- i ω (T + t_{0})} - i f (ω) \int_{0}^{T} e^{- i ω (T - s)} c (s) d s .

Displayed formulas in this passage

unlabeled · line 1318
$h = L^{2} (R_{+}), (h ψ) (ω) = ω ψ (ω), f (ω) = 2 e^{- ω} .$
unlabeled · line 1325
$ψ (0, ω) = e^{- i ω t_{0}} f (ω), B (0) = 0.$
unlabeled · line 1331
$K (u) = (h + v (u) ∣ f ⟩ ⟨ f ∣ κ (u) ⟨ f ∣ κ (u) ∣ f ⟩ 1)$
unlabeled · line 1343
$A (u) = ⟨ f, ψ (u)⟩, c (u) = v (u) A (u) + κ (u) B (u) .$
unlabeled · line 1348
$ψ (T, ω) = f (ω) e^{- i ω (T + t_{0})} - i f (ω) \int_{0}^{T} e^{- i ω (T - s)} c (s) d s .$

Continuum interpretation and endpoint attainability

Displayed formulas in this passage

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