The domain is (where $ac$ = absolutely continuous)
$$D(P) = \{f_{ac} \in L^2(\mathbb{R}) | f' \in L^2(\mathbb{R})\}$$
The text I am studying says that the functions that belong to $D(P)$ vanish at infinity. Why? Assuming that affirmation, the domain of the self-adjoint operator should be smaller than $D(P)$ because is only necessary that $g$ and $g'$ are in $L^2(\mathbb{R})$ and so $P$ should not be self-adjoint. Where have I gone wrong?