Setting $n=k+l$ and using

$$\begin{aligned}
J_{k+l}(\alpha+\beta) = \sum_{l=-\infty}^{\infty}
J_l(\beta)J_k(\alpha) ~,\label{due}\end{aligned}$$

Eq. (\[ffft5\]) can be re-written as

$$\begin{aligned}
\bar{\rho}(\Omega) && =  \sum_{m,n} \bar{f}_{\zeta}(n+m-\Omega)
J_n\left[-\Omega A_1(B_1+B_2)+m A_1 B_1\right] J_m\left[-\Omega A_2
B_2\right] \cr && \times \exp[(\Omega  (B_1+B_2)-m  B_1)^2/2] ~.
\label{ffft6}\end{aligned}$$

We now apply the adiabatic approximation imposing that the width of the peaks in $\bar{f}_{\zeta}$ is much narrower than the harmonic separation $\omega_l$ between peaks. Analysis of Eq. (\[ffft6\]) and Eq. (\[fbarz\]) shows that due to the adiabatic approximation, the contribution to $\bar{f}(\Omega)$ for a given value of $m+n$, is peaked around $\Omega \simeq m+n$. This means that the terms in the sum over $m$ in Eq. (\[ffft6\]) can be analyzed separately for a fixed value of $m+n$, and one obtains

$$\begin{aligned}
\bar{\rho}(\Omega,m+n) && =  \sum_{n} \bar{f}_{\zeta}(n+m-\Omega)
J_n\left[-\Omega A_1(B_1+B_2)+m A_1 B_1\right] J_m\left[-\Omega A_2
B_2\right] \cr && \times \exp[(\Omega  (B_1+B_2)-m  B_1)^2/2] ~.
\label{ffft6sep}\end{aligned}$$

It should be remarked that due to the adiabatic approximation, and to non-resonant behavior of Bessel functions, in Eq. (\[ffft6\]) we can replace $\Omega$ with $m+n$ under the Bessel functions. In this way, $\bar{f}_\zeta$ can be interpreted as the Fourier transform of the electron bunch density. The physical meaning of all this, is that $\bar{f}_{\zeta}$ is peaked at frequencies $\Omega$ near 