For any $f\in L^2(\mathbb{R}^d)$ prove \begin{align}\left\lVert \int_{\mathbb{R}^d} e^{i |x-y|^2}f(y) dy-\int_{\mathbb{R}^d}e^{i |x-y|^2} e^{-|y|^2/a}f(y) dy \right\rVert_{L^2} \rightarrow 0\ \ \ \text{as } \ a\rightarrow \infty \end{align}
I can easily prove pointwise convergence using dominated convergence with $|f(y)|$ as upper bound. Is there some sort of extension to dominated convergence, that would prove the $L^2$ convergence above?