Consider the following oscillatory integral with two parameters: $$I(a,b)=\int_{\mathbb{R}}e^{i(ax^2+bx)}\psi(x)\,dx$$ where $\psi$ is smooth and supported in $\{x:|x|\in[1/2,2]\}$.

Can we write $I(a,b)$ as $f(a)g(b)$ for some functions $f$ and $g$? i.e. we want to separate $a$ and $b$. I guess we have to find an asymptotic formula for $I(a,b)$ first but I do not know how to deal with the two parameters oscillatory integral (If there were only one parameter $\lambda$, then I can possibly use the *Van der corput* Lemma).