A note on the Fröhlich dynamics in the strong coupling limit

Abstract

We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in [Gri17]. In the latter it was shown that the Fröhlich time evolution applied to the initial state ${\phi }_0\otimes {\xi }_{\alpha }$, where ${\phi }_0$ 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.