four components of the electron four-momentum $p^{\mu}=(\varepsilon ,\bm{p})$, although the energy is a function of the linear momentum). Following the general notation in Ref. [@Landau_b_4_1982], these states can be written as $U_s(p,x)=E(p,x)u_s(p)$, where $$\label{E_p}
E(p,x)=\bigg[1+\frac{e\hat{n}\hat{A}(\phi)}{2p_-}\bigg]\text{e}^{i\left\{-(px)-\int_{-\infty}^{\phi}d\varphi\left[\frac{e(pA(\varphi))}{p_-}-\frac{e^2A^2(\varphi)}{2p_-}\right]\right\}},$$ and where $u_s(p)$ are the free, positive-energy spinors normalized as $u^{\dag}_s(p)u_{s'}(p)=2\varepsilon \delta_{ss'}$ [@Landau_b_4_1982]. In Eq. (\[E\_p\]) we have introduced the notation $\hat{v}=\gamma^{\mu}v_{\mu}$ for a generic four-vector $v^{\mu}$, with $\gamma^{\mu}$ being the Dirac matrices, which satisfy the anti-commutation relations $\{\gamma^{\mu},\gamma^{\nu}\}=2\eta^{\mu\nu}$ [@Landau_b_4_1982].

The electron Green’s function $G(x,x')$ in the general plane-wave background electromagnetic field described by the four-vector potential $A^{\mu}(\phi)$ is defined by the equation $$\{\gamma^{\mu}[i\partial_{\mu}-eA_{\mu}(\phi)]-m\}G(x,x')=\delta^{(4)}(x-x').$$ In order to uniquely identify the Green’s function, boundary conditions have also to be specified. Here, we always assume the Feynman prescription corresponding to the shift $m\to m-i0$ [@Landau_b_4_1982]. Within the operator technique the operator $G$ corresponding to the Green’s function $G(x,x')$ is defined via the equation $G(x,x')=\langle x|G|x'\rangle$, i.e., as $$G=\frac{1}{\hat{\Pi}-m+i0},$$ where $\Pi^{\mu}=P^{\mu}-eA^{\mu}(\Phi)$. Now, we have explicitly shown in Ref. [@Di_Piazza_2018_d] (see also Refs. [@Baier_1976_a; @Baier_1976_b; @Di_Piazza_2007]) that the operator $G$ can be written in the form $$\label{G_1}
\begin{split}
G&=(\hat{\Pi}+m)\frac{1}{\hat{\Pi}^2-m^2+i0}=(\hat{\Pi}+m)(-i)\int_0^{\infty}ds\, e^{-im^2s}e^{2isP_TP_{\phi}}\\
&\times e^{-i\int_0^sds'[\bm{P}_{\perp}-e\bm{A}_{\perp}(\Phi-2s'P_T)]^2}\Big\{1-\frac{e}{2P_T}\hat{n}[\hat{A}(\Phi-2sP_T)-\hat{A}(\Phi)]\Big\},
\end{split}$$ where the prescription $m^2\to m^2-i0$ is understood. Below, we will also need the equivalent expression $$\label{G_2}
