论文标题
关于FröhlichDynamics在强耦合极限中的注释
A note on the Fröhlich dynamics in the strong coupling limit
论文作者
论文摘要
我们在[GRI17]中获得的强耦合极限中对Fröhlich动力学进行了先前的结果。在后者中，结果表明，fröhlich时间的演变应用于初始状态$φ_0\ otimesξ_α$，其中$φ_0$是Pekar Energy函数的电子基态，$ξ_α$可以通过与$α^2 $ $α^2 $相比的全球阶段相关的声音相关的声音相关状态。在本说明中，我们证明，如果一个与$α^{ - 2} $成正比生成的声子，则类似的近似值适用于$ t = o（α^2）$，并且在creation和creation and quartion and quadratic中产生了一个非平凡的有效动力。我们的结果表明，电子基态在订单$α^2 $时保持接近其初始状态，而相干状态$ξ_α$周围的声子波动可以用时间依赖性的bogoliubov变换来描述。
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.
