0} 
\frac{e^{-\pi\omega\lambda}}{2\pi\omega\lambda} 
\delta(\varepsilon) \int_{x_0 \omega }^{\infty} 
y^{-\mathrm{i}\omega\lambda} e^{- y} \mbox{d}y \; 
\int_{x_0 \omega }^{\infty} \tilde{y}^{\mathrm{i}\omega\lambda} e^{-\tilde{y}} 
\mbox{d}\tilde{y} \ , \end{aligned}$$ with the differentiation on the $\delta$-function interpreted as $0$.[^2] If we now put all pieces together, we finally find for the vacuum expectation value of the number density operator $N(\omega)$ \[in case (1)\]: $$\begin{aligned}
\int_0^{\infty} \vert \beta^1_{\omega'\omega} \vert^2 \; \mbox{d}\omega' &=&
\frac{e^{-\pi\omega \lambda}}{2\pi \lambda \omega } \delta(\varepsilon) 
\vert \Gamma(1-\mathrm{i}\omega\lambda,x_0 \omega)\vert^2
\; \; 
+ \mbox{finite} \nonumber \\ \label{erg1}
\Rightarrow \frac{N(\omega)}{{\bf{V}}}
&=&\frac{e^{-\pi\omega \lambda}}{2 \pi \lambda \omega } 
\vert \Gamma(1-\mathrm{i}\omega\lambda,x_0 \omega)\vert^2
\; \; .\end{aligned}$$ The spectral energy flux through a 2-dimensional surface $\rho(\omega)$ then reads $$\begin{aligned}
 \label{energdi1}
\rho(\omega)&=& \frac{e^{-\pi\omega \lambda}\omega^2}{4\pi^3\lambda} 
\vert \Gamma(1+\mathrm{i}\omega\lambda,x_0 \omega)\vert^2\; .\end{aligned}$$

We have made several assumptions, but some of them may be relaxed. First, we assumed that the velocity $\nu$ of the mass shell is constant. In a realistic scenario we would expect a high velocity at the beginning, decreasing towards a constant value (inertial motion), in contrast to a collapse where the velocity would increase. Secondly, we approximated around the space-time points that cause the maximal amount of particle production. Therefore, our result is only correct in the vicinity of $u=0$. To obtain a result for later times, we 