Let $\phi(x) = a(x)\exp\left({i \cfrac{\xi_0.x}{h}}\right) $, with $\xi_0 \in \mathbb R^d$ fixed and $a \in C^{\infty}_c(\mathbb R^d)$. Let $f \in C^{\infty}_c(\mathbb R^{2d}) $. I want to show that $$ (Op^W_h(f)\phi)(x) = b(x,h)\exp\left({i \cfrac{\xi_0.x}{h}}\right)$$ with $b$ verifying an asymptotic expansion, $$ b(x,h) \sim \sum_{j=0}^{\infty}h^jb_j(x),$$ and explicitly find $b_0$ and $b_1$ ($Op^W_h$ being the Weyl quantization). The exercise says to use the quadratic stationnary phase lemma, but I don’t know how to do it. Let me precise what we have here.
$$ (Op^W_h(f)\phi)(x) = (2\pi h)^{-d} \int \int \exp\left({i \cfrac{(x-y).\eta}{h}}\right) f\left(\cfrac{x-y}{2},\eta\right)\phi(y)dyd\eta,$$ so we get
$$ (Op^W_h(f)\phi)(x) = (2\pi h)^{-d} \int \int \exp\left({i \cfrac{(x-y).\eta}{h}}\right) f\left(\cfrac{x-y}{2},\eta\right)a(y)\exp\left({i \cfrac{\xi_0.y}{h}}\right)dyd\eta.$$
Could someone please explain at what element in this double integral I should apply the stationary phase lemma, which is for integrals looking like $$\int c(x)\exp\left({i \cfrac{\psi(x)}{h}}\right)dx $$ and why « quadratic » ? Any suggestion would be nice, thanks.