Advertisement

Letters in Mathematical Physics

, Volume 109, Issue 7, pp 1701–1702 | Cite as

Correction to: Completely positive dynamical semigroups and quantum resonance theory

  • Martin KönenbergEmail author
  • Marco Merkli
Correction
  • 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 Xwhich 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)
The corrected bound (0.2) yields (0.1) by the argument in the paper. We get (0.2) as follows,
$$\begin{aligned} \Big | \mathrm{e}^{\mathrm{i}t(e +\lambda ^2 a_{e,j})} - \mathrm{e}^{\mathrm{i}t ({{\widetilde{e}}} +\lambda ^2 {{\widetilde{\lambda }}}_{{{\widetilde{e}}},j}) } \Big |= & {} \mathrm{e}^{-t \lambda ^2 {\mathrm{Im}}a_{e,j}} \ \Big | 1- \mathrm{e}^{\mathrm{i}t ({{\widetilde{e}}}-e +\lambda ^2 (\widetilde{\lambda }_{{{\widetilde{e}}},j} - a_{e,j}) )} \Big | \\= & {} \mathrm{e}^{-t \lambda ^2 {\mathrm{Im}}a_{e,j}} \ \Big | \int _{0}^{ t({{\widetilde{e}}} - e +\lambda ^2 ( {{\widetilde{\lambda }}}_{{{\widetilde{e}}},j} - a_{e,j}))} \mathrm{e}^{\mathrm{i}z} \hbox {d}z \Big | \\\le & {} \mathrm{e}^{-t \lambda ^2 {\mathrm{Im}}a_{e,j}} \ t \big | {{\widetilde{e}}} - e +\lambda ^2 ( {{\widetilde{\lambda }}}_{{{\widetilde{e}}},j} - a_{e,j})\big |\ \mathrm{e}^{t \lambda ^2 |{\mathrm{Im}} ( {{\widetilde{\lambda }}}_{{{\widetilde{e}}},j} - a_{e,j} ) |}. \end{aligned}$$
Using in the last inequality that \(\mathrm{Im}a_{e,j}=(1+O(\lambda ))\gamma \), that \({{\widetilde{\lambda }}}_{{{\widetilde{e}}},j}-a_{e,j}=O(\lambda )\) and that
$$\begin{aligned} {{\widetilde{e}}} - e=O(\lambda ^2) \end{aligned}$$
(0.3)
gives (0.2). We note that by Lemma 3.1 in the published paper, the bound (0.3) above is seemingly only \(O(\lambda )\). However, since \(\rho _{\mathrm{S},\beta ,\lambda }-\rho _{\mathrm{S},\beta ,0} = O(\lambda ^2)\) (the linear term vanishes since the interaction has vanishing average in the reservoir vacuum state), we have \({\widetilde{H}}_\mathrm{S}-H_\mathrm{S}=O(\lambda ^2)\). The \(O(\lambda )\) bounds in Lemma 3.1 are thus actually \(O(\lambda ^2)\) bounds and (0.3) holds.

Notes

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMemorial UniversitySt. John’sCanada
  2. 2.Fachbereich MathematikUniversität StuttgartStuttgartGermany

Personalised recommendations