Manuscript Em05-B@1 · Em05-B-v1.tex · proposition · 6c050f5b

Proposition: No exact finite-time continuum transfer

A declared continuum model and endpoint non-attainability · lines 2266–2302 · proof included or linked

Em05-B@1#prop:continuumEndpoint

In the model (eq:continuumGenerator),

∣ B (T) ∣ < 1

for every finite $T$ .

Proof. If $∣ B (T) ∣ = 1$ , unitarity and the normalized initial state force $ψ (T) = 0$ . Since $f (ω) > 0$ , (eq:packetFormula) implies

e^{- i ω t_{0}} = i \int_{0}^{T} e^{i ω s} c (s) d s

for almost every positive $ω$ .

The bounded function $c$ is integrable on $[0, T]$ . Both sides extend to entire functions of complex $ω$ . Their equality almost everywhere on a real interval, together with continuity and the identity theorem, gives equality everywhere.

At $ω = i y$ , $y > 0$ , the left side is $e^{y t_{0}}$ , whereas

\int_{0}^{T} e^{- y s} c (s) d s \leq \frac{∥ c ∥ _{L^{\infty}}}{y} .

The right side tends to zero, while the left side is at least one. This is a contradiction.

Displayed formulas in this passage

unlabeled · line 2269
$∣ B (T) ∣ < 1$
unlabeled · line 2279
$e^{- i ω t_{0}} = i \int_{0}^{T} e^{i ω s} c (s) d s$
unlabeled · line 2294
$\int_{0}^{T} e^{- y s} c (s) d s \leq \frac{∥ c ∥ _{L^{\infty}}}{y} .$

Proposition: No exact finite-time continuum transfer

Displayed formulas in this passage

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