deletions | additions       diff --git a/photoemission2.tex b/photoemission2.tex  index 4dd71ba..996b9be 100644  --- a/photoemission2.tex  +++ b/photoemission2.tex 
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\begin{equation}  P(\vec{k})=\lim_{t\rightarrow \infty}\sum_i^N |\phi^B_i(\vec{k},t)|^2\,,  \end{equation}  while the energy-resolved probability follows by direct integration integration,  $P(E)=\int_{E=|\vec{k}|^2/2}{\rm d}\vec{k}P(\vec{k})$.   In Eq.~\eqref{eq:FMM_prop_aux} we introduced the Volkov propagator $\hat{U}_{\rm v}(\Delta t)$ for the wavefunctions  in $B$.   It is the time-evolution operator associated with the Hamiltonian $\hat{H}_{\rm v}$ describing free electrons in   an oscillating field.   Given a time dependent vector field $\vec{A}(t)$, $\vec{\mathrm{A}}(t)$,  the Hamiltonian $\hat{H}_{\rm v}=\frac{1}{2}(-i\vec{\nabla}-\frac{\vec{A}(t)}{c})^2$ v}=\frac{1}{2}(-i\vec{\nabla}-\frac{\vec{\mathrm{A}}(t)}{c})^2$  expressed in the velocity gauge is diagonal in momentum and can be naturally applied to $\phi^B_i(\vec{k},t)$.  For all systems that can be described by a Hamiltonian such that $\hat{H}(\vec{r},t)=\hat{H}_{\rm v}(\vec{r},t)$   for $\vec{r} \in B$ and all times time  $t$, Eqs. eqs.  \eqref{eq:FMM_prop} and \eqref{eq:FMM_prop_aux} are equivalent to a time propagation in the entire space   $A\cup B$.  In particular, it exactly describes situations where the electrons follow trajectories crossing the boundary separating $A$ and $B$ as illustrated in Fig.~\ref{fig:pes_sheme}(b).