we use (A.29c) to first absorb the term $(\mathrm{Re} \, p_1(\omega)) t = \mathcal{O}(\omega^{5/2})$ (uniformly with respect to $t$) by half of $\mathrm{Re} \, p_0(\omega)$. We can then absorb $(\mathrm{Re} \, p_3(\omega)) t^3$ on the considered interval by part of $(\mathrm{Re} \, p_2(\omega)) t^2$ (this argument was also used in the proof of Proposition A.3). The final term $(\mathrm{Re} \, p_4(\omega)) t^4$ is $\mathcal{O}(\omega t^2)$ so it can also be absorbed by half of what is left from $(\mathrm{Re} \, p_2(\omega)) t^2$ (up to choosing $\omega$ small enough). In the end we get an estimate (with uniform constants $C$ and $c$):

$$|\mathcal{H}_b(x, y)| \le C \omega^{p/2} e^{-c \omega^2 y} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2c_4}}{9c_3^2} \omega}^{\sqrt{\frac{2\omega}{3c_3}}} \exp(-c\omega^{1/2}y t^2) dt.$$

Estimating crudely the integral over the segment by the integral of the same function over $\mathbb{R}$, we end up with:

$$|\mathcal{H}_b(x, y)| \le C \frac{\omega^{p/2-1/4}}{y^{1/2}} e^{-c \omega^2 y},$$

and the same estimate holds for $\mathcal{H}_b(x, y)$ that is the complex conjugate of $\mathcal{H}_b(x, y)$.

Let us assume that $p$ is odd. Combining the latter estimate with (A.58), we get:

$$|\mathbf{G}_p(x, y)| \le C \frac{\omega^{p/2-1/4}}{y^{1/2}} e^{-c \omega^2 y} + C e^{-c \omega^{3/2} y} \left( \frac{1}{\omega^{(p+1)/4} y^{(p+1)/2}} + \frac{\omega^{p/2}}{\omega y} \right).$$

Since $\omega$ has been chosen smaller than some constant $\omega_0$, there is no loss of generality in assuming $\omega_0 \le 1$ and we therefore have:

$$e^{-c \omega^{3/2} y} \le e^{-c \omega^2 y}.$$

We recall the definition $\omega = |x|/y$ and rewrite the latter estimate in terms of $x$ and $y$ to obtain:

$$|\mathbf{G}_p(x, y)| \le C \exp \left( -c \frac{x^2}{y} \right) \left\{ \left( \frac{|x|}{\sqrt{y}} \right)^{p/2} \frac{1}{|x|^{1/4} y^{(p+1)/4}} + \frac{1}{|x|^{(p+1)/4} y^{(p+1)/4}} + \left( \frac{|x|}{\sqrt{y}} \right)^{p/2} \frac{1}{|x| y^{p/4}} \right\}.$$

We can absorb all polynomial expressions of $|x|/\sqrt{y}$ by the Gaussian function and we also use the inequalities:

$$|x| \ge |x|^{1/4} y^{1/4}, \quad |x|^{p/4} \ge y_{\min}^{p/12} > 0,$$

that hold for $|x| \ge y^{1/3}$ and $y \ge y_{\min}$. Eventually, we have shown that there exists some small constant $c_b > 0$ and some constants $C$ and $c$ such that there holds:

$$\forall y \ge y_{\min}, \quad \forall x \in [-c_b y, -y^{1/3}], \quad |\mathbf{G}_p(x, y)| \le \frac{C}{|x|^{1/4} y^{(p+1)/4}} \exp \left( -c \frac{x^2}{y} \right). \quad (A.60)$$

The same kind of arguments lead to the estimate (A.60) in the case where $p$ is even (starting now from (A.59)).

• Step 5: the oscillating side. Part II. There is no real difficulty in adapting the proof of Proposition A.4 to this slightly more general framework that incorporates the factor $(\mathbf{i}\theta)^p$ in the definition (A.49) of $\mathbf{G}_p$. By following the same arguments as in the proof of Proposition A.4 and absorbing polynomial terms by exponentially decaying ones, we can show that there exists a constant $c_b > c_b$ and some

$$\forall y \ge y_{\min}, \quad \forall x \le -c_b y, \quad |\mathbf{G}_p(x, y)| \le \frac{C}{y^{(p+1)/4}} \exp \left( -c \frac{|x|^{4/3}}{y^{1/3}} \right). \quad (A.61)$$
• Step 6: the oscillating side. Part III. It remains to deals with the case $x \in [-C_b y, -c_b y]$ and this is done by merely adapting Proposition A.5. We choose again the contour depicted in Figure A.2 so that along the two inclined segments, the complex number $\theta$ is $\mathcal{O}(\delta)$ and the parameter $\delta$ is chosen such that it satisfies (A.37). This choice, that is uniform with respect to $x$ and $y$ in the considered regime, allows us to absorb the term $(i\theta)^p$ into a constant along the two inclined segments. For the integrals along the two horizontal half-lines, the polynomial factor $(i\theta)^p$ simply gives an algebraic factor that is harmless when compared with the exponentially decaying term $\exp(-cy)$. Overall, we leave as an exercise to the interested reader to prove that for suitable constants $C$ and $c$, there holds:

$$ \forall y \ge y_{\min}, \quad \forall x \in [-C_b y, -c_b y], \quad |G_p(x, y)| \le \frac{C}{y^{(p+1)/4}} \exp(-cy). \quad (A.62) $$

We can then use the above estimates (A.60), (A.61), (A.62) and adapt the arguments of Corollaries A.2 and A.3 to show that for any given constant $c > 0$, there exist constants $C$ and $c$ such that the estimates corresponding to the last two cases of (A.50) are valid. This completes the proof of Theorem A.3. $\square$

*Proof of Corollary A.7.* From the definition (A.46) of $A_r$ and the definition (A.49) of the correctors $G_p$, we have$^7$:

$$ \frac{\partial A_r}{\partial y}(x, y) = -c_3 G_2(x, y) - c_4 G_3(x, y). $$

We now consider $n \in \mathbb{N}^*$ and $j_0 \in \mathbb{N}^*$ such that $j_0$ belongs to the segment $[n|\alpha_r|/2, n]$. We observe that the segment $[j_0/|\alpha_r|, n]$ is included in $[n/2, n/|\alpha_r|]$ so the mean value theorem gives the bound:

$$ \left| \mathbf{A}_r \left( -j_0 + n |\alpha_r|, \frac{j_0}{|\alpha_r|} \right) - \mathbf{A}_r (-j_0 + n |\alpha_r|, n) \right| \le C |n| |\alpha_r| - j_0| \sup_{y \in [n/2, n/|\alpha_r|]} |\mathbf{G}_2(-j_0 + n |\alpha_r|, y)| + C |n| |\alpha_r| - j_0| \sup_{y \in [n/2, n/|\alpha_r|]} |\mathbf{G}_3(-j_0 + n |\alpha_r|, y)|. \quad (A.63) $$

Corollary A.7 then follows by applying Theorem A.4, keeping in mind that $\alpha_r$ is negative (so the relevant constant $c_3$ is negative) and that the relevant values of $x$ and $y$ satisfy here $|x|/y \le c$ for some constant $c$ that only depends on $\alpha_r$ (this is because of the bounds on $j_0$ in terms of $n$). Actually, the most critical case arises in the right-hand side of (A.63) with the term:

$$ |n| |\alpha_r| - j_0| \sup_{y \in [n/2, n/|\alpha_r|]} |\mathbf{G}_2(-j_0 + n |\alpha_r|, y)|, $$

in the regime $n|\alpha_r| - j_0 \ge n^{1/3}$. We then use Theorem A.4 for $p = 2$ and obtain a bound of the form:

$$ C |n| |\alpha_r| - j_0| \frac{1}{|n| |\alpha_r| - j_0|^{1/4} n^{3/4}} \exp \left( -c \frac{|n| |\alpha_r| - j_0|^2}{n} \right), $$

so the Gaussian term can absorb the polynomial expression $|n| |\alpha_r| - j_0|/\sqrt{n}$ and this gives a bound:

$$ \frac{C}{|n| |\alpha_r| - j_0|^{1/4} n^{1/4}} \exp \left( -c \frac{|n| |\alpha_r| - j_0|^2}{n} \right), $$

just like we had in Corollary A.1 for the free Green's function. In all other regimes, the situation is more favorable and there are extra positive powers of $n$ that can even be omitted in the end. $\square$

<footnote>$^7$We first differentiate the definition (A.46) with respect to $y$ and then apply once again the Cauchy formula to let the abscissa $\eta$ tend to zero.</footnote>
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