such that $F_N(z)={\mathrm{O}}({\mathrm{e}}^{-s})$ on $[2s_1,\infty)$. In the second part, we will just have to consider the case $s\in [s_0,2s_1]$, and show that $F_N$ is merely uniformly bounded on this interval.

### Case $\mathbf{ s\geq 0} $ 

In order to use the results linking parabolic cylinder functions and the Airy function (proved in @olver59 and cited in @olver75), let us define yet another auxiliary variable, $\eta$, by $$\frac{2}{3}\eta^{3/2}\beta^2=\int_{x_2}^xg^{1/2}(t)dt \;.$$

Then, if we call ${{\cal E}}$ and ${{\cal M}}$ the weight and modulus functions associated with the Airy function, we have, as shown in **A3**: $$\begin{aligned}
{\mathbf{E}^{-1}}(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda}) &\leq
{{\cal E}^{-1}}(\lambda^{2/3}\beta^{4/3}\eta)(1+{\mathrm{O}}((\lambda\beta^2)^{-1})) \; ,\\
{\mathbf{M}}(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda})&\leq
\frac{\sqrt{2}\pi^{1/4}\left(\Gamma((1+\lambda\beta^2)/2)\right)^{1/2}
\beta^{1/2}}{(\lambda\beta^2)^{1/12}}\\
&\times\left(\frac{\eta}{{\upsilon}^2-\beta^2}\right)^{1/4}{{\cal M}}(\lambda^{2/3}\beta^{4/3}\eta)
\left(1+{\mathrm{O}}((\lambda\beta^2)^{-1})\right) \; .\end{aligned}$$ Whence, if we call $\theta\triangleq \lambda^{2/3}\beta^{4/3}\eta$, $$|F_N(\lambda x)|\leq K_{n,N}x^{1/2}(\eta/(x^2-4lx+4))^{1/4}{{\cal E}^{-1}}(\theta){{\cal M}}(\theta)
\left(1+{\mathrm{O}}((\lambda\beta^2)^{-1}\vee \lambda^{-2/3})\right) \; .$$ In **A4**, we show that $K_{n,N}\sim 2^{2/3}(N/n)^{1/4}$ under ${\mathbf{AB}}$. From now on, $\Delta$ will denote a generic constant; its value may change from display to display. As long as $x\geq x_2$, or $s\geq 0$, we have $$|F_N(\lambda x)|\leq \Delta (N/n)^{1/4} x^{1/2}(\eta/(x^2-4lx+4))^{1/4}{{\cal E}^{-1}}(\theta){{\cal M}}(\theta)) \; .$$ Now using the fact that (see @olver, chap. 11) $x^{1/4}{{\cal M}}(x)\leq \Delta$, ${{\cal E}^{-1}}(x)\leq \Delta
\exp(-2x^{3/2}/3)$ for $x\geq 0$ and $\lambda\beta^2=2N+1$, we get the new inequality $$|F_N(\lambda x)|\leq \Delta
\left(\frac{N}{n}\right)^{1/4}N^{-1/6}\left(\frac{x^2}{x^2-4lx+4}\right)^{1/4}\exp(-(2\theta^{3/2})/3) \; .$$ 