A note on the Fröhlich dynamics in the strong coupling limit

Abstract

AbstractWe 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 $$\varphi _0 \otimes \xi _\alpha $$ φ 0 ⊗ ξ α , where $$\varphi _0$$ φ 0 is the electron ground state of the Pekar energy functional and $$\xi _\alpha $$ ξ α the associated coherent state of the phonons, can be approximated by a global phase for times small compared to $$\alpha ^2$$ α 2 . In the present note we prove that a similar approximation holds for $$t=O(\alpha ^2)$$ t = O ( α 2 ) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to $$\alpha ^{-2}$$ 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 $$\alpha ^2$$ α 2 , while the phonon fluctuations around the coherent state $$\xi _\alpha $$ ξ α can be described by a time-dependent Bogoliubov transformation.