On the dynamics of the mean-field polaron in the high-frequency limit (Q2410936)

The authors study the dynamics of the mean-field polaron in the limit of infinite phonon frequency $\omega\rightarrow\infty$, which is described by the coupled equations \begin{align*} i\partial_{t}u & =-\Delta u+vu,\tag{1}\\ \varepsilon^{2}\partial_{t}^{2}v & =-v+\Delta^{-1}\left\vert u\right\vert^{2},\tag{2} \end{align*} for the wave function $u=u(x,t)\in\mathbb{C}$ of the electron and the electrostatic potential $v=v(x,t)\in\mathbb{R}$. The aim of the paper is to compare the above system with the Choquard equation \[ i\partial_{t}U=-\Delta U+\Delta^{-1}\left\vert U\right\vert ^{2}U,\tag{3} \] a Hartree-type equation obtained by formally passing to the limit $\varepsilon\rightarrow0$ in (1)--(2). The authors give sufficient conditions to assure that for all sufficiently small $\varepsilon>0$, the solutions to (1)--(2) can be approximated by solutions to the Choquard equation (3), which means that (3) makes correct predictions about the dynamics of the polaron mean-field model for small values of $\varepsilon=1/\omega$. The result is obtained by exploiting the corresponding integral equation and the highly oscillatory linear semigroup associated with (2) to get rid of the $\varepsilon^{-1}$ singularity in front of the nonlinear terms.