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 $φ_0 \otimes ξ_α$, where $φ_0$ is the electron ground state of the Pekar energy functional and $ξ_α$ the associated coherent state of the phonons, can be approximated by a global phase for times small compared to $α^2$. In the present note we prove that a similar approximation holds for $t=O(α^2)$ if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to $α^{-2}$ 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 $α^2$ while the phonon fluctuations around the coherent state $ξ_α$ can be described by a time-dependent Bogoliubov transformation.