# Correction to: Completely positive dynamical semigroups and quantum resonance theory

Correction

First Online:

- 171 Downloads

## 1 Correction to: Lett Math Phys (2017) 107:1215–1233 https://doi.org/10.1007/s11005-017-0937-z

**Correction.**The bound (2.23) in Theorem 2.1 has to be replaced by

$$\begin{aligned} \big | R_{\lambda ,t}(X)\big | \le C\, \big ( |\lambda | + \lambda ^2 t \big )\ \mathrm{e}^{-\lambda ^2(1+O(\lambda ))\gamma t} \ \Vert X\Vert . \end{aligned}$$

(0.1)

**Implication.**The difference is that in reality we can only show \(\lambda ^2t\) on the right side, instead of the \(|\lambda |^3t\) as announced in the published paper. The remainder (0.1) is still asymptotically exact (vanishing as \(t\rightarrow \infty \)), and our result still proves that the dynamics is approximated, asymptotically exactly, by a CPT semigroup. But for times \(t\approx 1/\lambda ^2\), the approximation is not guaranteed to be small.

Nevertheless, as will be discussed elsewhere in more detail, we can obtain a proof of Theorem 2.1, exactly as stated in the published paper, if instead of allowing all observables \(X\in {\mathcal B}({{\mathcal {H}}}_\mathrm{S})\), we restrict to *X**which commute with*\(H_\mathrm{S}\). Such observables determine the dynamics of the system populations (diagonal elements of the reduced density matrix). This means that the true population dynamics is approximated by a CPT semigroup dynamics, uniformly in time to accuracy \(O(\lambda )\), and the approximating Markovian dynamics is also asymptotically exact.

**Proof of the correction**(0.1). An estimate for the remainder term in the proof of Theorem 2.1, as given in the paper, contains a mistake. Namely, in footnote 5 [after Eq. (3.44)], the real phase of the exponential was omitted. The correct estimate is

$$\begin{aligned} \big | \mathrm{e}^{\mathrm{i}t(e +\lambda ^2 a_{e,j})} - \mathrm{e}^{\mathrm{i}t ({{\widetilde{e}}} +\lambda ^2 {{\widetilde{\lambda }}}_{{{\widetilde{e}}},j}) }\big | \le C \lambda ^2 t\ \mathrm{e}^{-\lambda ^2(1+O(\lambda ))\gamma t}. \end{aligned}$$

(0.2)

$$\begin{aligned} \Big | \mathrm{e}^{\mathrm{i}t(e +\lambda ^2 a_{e,j})} - \mathrm{e}^{\mathrm{i}t ({{\widetilde{e}}} +\lambda ^2 {{\widetilde{\lambda }}}_{{{\widetilde{e}}},j}) } \Big |= & {} \mathrm{e}^{-t \lambda ^2 {\mathrm{Im}}a_{e,j}} \ \Big | 1- \mathrm{e}^{\mathrm{i}t ({{\widetilde{e}}}-e +\lambda ^2 (\widetilde{\lambda }_{{{\widetilde{e}}},j} - a_{e,j}) )} \Big | \\= & {} \mathrm{e}^{-t \lambda ^2 {\mathrm{Im}}a_{e,j}} \ \Big | \int _{0}^{ t({{\widetilde{e}}} - e +\lambda ^2 ( {{\widetilde{\lambda }}}_{{{\widetilde{e}}},j} - a_{e,j}))} \mathrm{e}^{\mathrm{i}z} \hbox {d}z \Big | \\\le & {} \mathrm{e}^{-t \lambda ^2 {\mathrm{Im}}a_{e,j}} \ t \big | {{\widetilde{e}}} - e +\lambda ^2 ( {{\widetilde{\lambda }}}_{{{\widetilde{e}}},j} - a_{e,j})\big |\ \mathrm{e}^{t \lambda ^2 |{\mathrm{Im}} ( {{\widetilde{\lambda }}}_{{{\widetilde{e}}},j} - a_{e,j} ) |}. \end{aligned}$$

$$\begin{aligned} {{\widetilde{e}}} - e=O(\lambda ^2) \end{aligned}$$

(0.3)

## Notes

## Copyright information

© Springer Nature B.V. 2019