Recall for every $f \in L^2\left(\mathbb{R}^3\right)$, one has that $g(x):=(-\Delta+1)^{-1} f$ is a well-defined $L^2\left(\mathbb{R}^3\right)$ function. A nd one may compute $g$ by solving
$$
(-\Delta+1) g=f
$$
(Recall $\Delta$ in $\mathbb{R}^3$ is defined as $\Delta:=\sum_{i=1}^3 \partial_i^2$, also recall one may also define $(-\Delta+1)^{-1}$ by Fourier theory.)

Now, let $V(x):=e^{-|x|^2}, x \in \mathbb{R}^3$. Prove that the operator $T:=I+(-\Delta+1)^{-1} V$ is invertible in $L^2$.
(Here, $T f:=f+(-\Delta+1)^{-1}(V f)$.)