Consider the class of function defined by $$\mathcal{G}=\operatorname{Span}\left\{e^{-\frac{(x+a)^2}{2}}-e^{-x}e^{-\frac{(x+a)^2}{2}}\mid a\in\mathbb{R}\right\}.$$ Is $\mathcal{G}$ dense in $L^2(\mathbb{R}, e^x\,dx)$?
Note: If in the expression of $\mathcal{G}$, we had $e^{-(x+a)}$ instead of $e^{-x}$, the result would be true (by Wiener's Tauberian theorem).
Edit: Here is a motivation why I need a result of this sort. I am working with a differential operator $A$ on $L^2(\mathbb{R}, e^x\,dx)$ such that $$Au(x)=e^{-x}u''(x)$$ for all $u\in C^2_b(\mathbb{R})\cap L^2(\mathbb{R},e^x\,dx)$. It can be shown that $A$ is the generator of a Markov diffusion semigroup. I need to show that for any $\alpha>0$, $R^{-1}_\alpha(\mathcal{G})$ is dense in $L^2(\mathbb{R}, e^x\,dx)$, where $R_\alpha=(\alpha-A)^{-1}$ is the resolvent operator.