the [parabolic cylinder functions]{} results in terms of Airy functions and derive the elements we need to complete the proof of (\[whattoshow1\]) and (\[whattoshow2\]).

The reason for which we could not exactly follow the “original" method is related to the error control function called ${\cal V}(\zeta)$ in @imj. This function depends upon the parameter $\omega=2\lambda/\kappa$, which in the case $n/N{\rightarrow}\gamma \in \mathbb{R}$ is bounded away from 2. This essentially allows a uniform control over ${\cal V}$, and it is possible to show that this error control function is bounded as a function of $N$. Since the control is actually something like $\exp(\lambda_0 {\cal V}/\kappa)-1$, it tends to zero as $N{\rightarrow}\infty$. This gave @imj a way to get part of (\[whattoshow2\]).\
In our case, it seems that ${\cal V}$ would tend to $\infty$, at a rate that is nevertheless ${\mathrm{o}}(\kappa)$. As it seems easier and more promising to use @olver80 than to derive the growth of ${\cal V}$, we choose this approach. Nevertheless, this is the only (but crucial) technicality (in the complex case) that did not carry through by the method described in @imj under ${\mathbf{AB}}$.

### Real Case

For the $c_{\phi}$ problem, we provide a closed form expression at 