the domain is: $$D(P) = \{f \in C^{\infty}(\mathbb{R}) \cap L^2(\mathbb{R})\}$$ I think that is because some functions that belong to $D(P)$ should not be bounded at infinity.
The text I am studying suggests an example for bounded intervals, assuming that the domain of $P$, in this case, is: $$D(P) = \{f \in C^{\infty}([a,b]) \cap L^2([a,b])\}$$ an example of the non-boundness of $P$ is represented by: $f_n(x) = e^{-nx}$ in fact: $$\frac{|| Pf_n(x)||}{||f_n||} = in \rightarrow \infty$$