We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state \(\varphi _0 \otimes \xi _\alpha \), where \(\varphi _0\) is the electron ground state of the Pekar energy functional and \(\xi _\alpha \) the associated coherent state of the phonons, can be approximated by a global phase for times small compared to \(\alpha ^2\). In the present note we prove that a similar approximation holds for \(t=O(\alpha ^2)\) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to \(\alpha ^{-2}\) and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order \(\alpha ^2\), while the phonon fluctuations around the coherent state \(\xi _\alpha \) can be described by a time-dependent Bogoliubov transformation.

我们修改了关于Griesemer中强耦合极限中Fröhlich动力学的先前结果（Rev Math Phys 29（10）：1750030，2017）。在后者中，表明Fröhlich时间演化适用于初始状态\（\ varphi _0 \ otimes \ xi _ \ alpha \），其中\（\ varphi _0 \）是Pekar能量泛函的电子基态。与（\ alpha ^ 2 \）相比，\（\ xi _ \ alpha \）声子的相关相干状态可以通过全局相位近似地相乘。在本说明中，我们证明如果\（t = O（\ alpha ^ 2）\）包含声子的非平凡有效动力学，且该近似动力学近似于\（t = O（\ alpha ^ 2）\），则该近似近似成立。\（\ alpha ^ {-2} \）并在创建和an灭运算符中为平方。我们的结果表明，电子基态在阶数\（\ alpha ^ 2 \）的时间内保持接近其初始状态，而围绕相干态\（\ xi _ \ alpha \）的声子涨落可以用一个时间来描述。依赖的Bogoliubov变换。