A note on the Frohlich dynamics in the strong coupling limit

We revise a previous result about the Frohlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Frohlich time evolution applied to the initial state phi(0) circle times xi(alpha), where phi(0) is the electron ground state of the Pekar energy functional and 4 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.