Manuscript En01-B@2 · En01-B-v2.tex · theorem · f226d1eb

Theorem: Bounded regularized kick generator

Declared regularized model and its dynamics › Convention and rigid-body specification · lines 1001–1057 · proof included or linked

En01-B@2#thm:kick, En01-B@2#eq:kick, En01-B@2#eq:rate

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Put $k_{e} = k \cdot e$ , $U_{k} = e^{i k_{e} x}$ , and

Q_{r_{c}} (d k) = \frac{G}{2 π ^{2} ℏ} \frac{∣ f _{r_{c}} ( k ) ∣ ^{2}}{∣ k ∣ ^{2}} d^{3} k .

Then $Q_{r_{c}}$ is a finite positive even measure and

G_{r_{c}} (ρ) = \int (U_{k} ρ U_{k}^{†} - ρ) Q_{r_{c}} (d k) .

Writing $q_{r_{c}} = Q_{r_{c}} (R^{3})$ ,

q_{r_{c}} \leq \frac{G M ^{2}}{π ℏ r _{c}}, ∥ G_{r_{c}} ∥_{1 \to 1} \leq 2 q_{r_{c}} .

Its exponential is a CPTP semigroup on trace-class operators. The position coherence rate is

Γ_{r_{c}} (d) = \int [1 - cos (k_{e} d)] Q_{r_{c}} (d k) .

Proof. Since $∣ μ (k) ∣ \leq M$ ,

q_{r_{c}} \leq \frac{2 G M ^{2}}{π ℏ} \int_{0}^{\infty} e^{- r_{c}^{2} k^{2}} d k = \frac{G M ^{2}}{π ℏ r _{c}} .

Fourier transforming $1/∣ r ∣$ gives $4 π /∣ k ∣^{2}$ . On smooth finite-rank position kernels, substitution in (eq:DP) gives multiplication by $- Γ_{r_{c}} (x - x^{'})$ . Evenness removes the sine part in (eq:kick). The bounded integral extends the identity to trace-class operators.

For $q_{r_{c}} > 0$ , define

J (ρ) = q_{r_{c}}^{- 1} \int U_{k} ρ U_{k}^{†} Q_{r_{c}} (d k) .

Then

e^{t G_{r_{c}}} = e^{- t q_{r_{c}}} n = 0 \sum^{\infty} \frac{( t q _{r_{c}} ) ^{n}}{n !} J^{n}

is a norm-convergent CPTP mixture. The zero-rate case is the identity. The norm bound follows from the norm-one unitary conjugations.

Displayed formulas in this passage

En01-B@2#eq:kick · line 1010
$G_{r_{c}} (ρ) = \int (U_{k} ρ U_{k}^{†} - ρ) Q_{r_{c}} (d k) .$
En01-B@2#eq:rate · line 1022
$Γ_{r_{c}} (d) = \int [1 - cos (k_{e} d)] Q_{r_{c}} (d k) .$
unlabeled · line 1004
$Q_{r_{c}} (d k) = \frac{G}{2 π ^{2} ℏ} \frac{∣ f _{r_{c}} ( k ) ∣ ^{2}}{∣ k ∣ ^{2}} d^{3} k .$
unlabeled · line 1015
$q_{r_{c}} \leq \frac{G M ^{2}}{π ℏ r _{c}}, ∥ G_{r_{c}} ∥_{1 \to 1} \leq 2 q_{r_{c}} .$
unlabeled · line 1030
$q_{r_{c}} \leq \frac{2 G M ^{2}}{π ℏ} \int_{0}^{\infty} e^{- r_{c}^{2} k^{2}} d k = \frac{G M ^{2}}{π ℏ r _{c}} .$
unlabeled · line 1044
$J (ρ) = q_{r_{c}}^{- 1} \int U_{k} ρ U_{k}^{†} Q_{r_{c}} (d k) .$
unlabeled · line 1049
$e^{t G_{r_{c}}} = e^{- t q_{r_{c}}} n = 0 \sum^{\infty} \frac{( t q _{r_{c}} ) ^{n}}{n !} J^{n}$

Theorem: Bounded regularized kick generator

Displayed formulas in this passage

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