The vacuum expectation value of the number density operator is given by $$\label{Ablx}
\int_0^{\infty}\vert\beta^1_{\omega'\omega}\vert^2 \; {\mathrm{d}}\omega' =
\frac{d_0^2}{4\pi^2 c_1^2} \int_0^{\infty} \frac{\omega'}{\omega} 
\Biggl|\int_{x_0}^{\infty} e^{-\mathrm{i}\omega_+ x} 
x^{-\mathrm{i}\omega \lambda} \; {\mathrm{d}}x\Biggr|^2\;{\mathrm{d}}\omega'\ .$$ This quantity is divergent and has to be regularized. First we will shift the $\omega'$-integration into the complex plane for some $\epsilon$. We do this because, for ${\rm Im}\, \omega_+ < 0$, the inner integrations are well defined and may be expressed as incomplete $\Gamma$-functions [@AS]. Further we regularize the number density with the help of a ‘frequency split’ $$\begin{aligned}
\vert \beta^1_{\omega'\omega} \vert^2 = \lim_{\omega \to \tilde{\omega}} 
\beta^1_{\omega'\omega} 
{\beta^1_{\tilde{\omega}\omega}}^*\ .\end{aligned}$$ Finally, we take the difference to the limit $x_0 \to 0$, which allows us to subtract the divergent part in a well defined manner. The point is that (\[Ablx\]) reduces in this limit to the expression for the number density that shows up in the Hawking effect [@Haw], $$\label{x0lim}
\lim_{x_0 \to 0}\int_0^{\infty} \vert\beta^1_{\omega'\omega}\vert^2 \;
{\mathrm{d}}\omega'  = \lim_{\tilde\omega \to \omega}
\frac{\delta(\tilde\omega - \omega)}{e^{2\pi\omega\lambda}-1}\ .$$ The only difference is that $\lambda$ replaces the inverse of the surface gravity, $1/\kappa = 4 M$, that would show up in the situation of a collapsing mass shell.

As a first step we rotate the integration variable 