Condition for vector to be in the domain of unbounded operator. Let $P$ be unbounded self-adjoint operator on some Hilbert space $\mathcal{H}$. We assume that the limit
$$
\lim_{\epsilon \searrow 0} \|\exp(-\epsilon^2 P^2/2) P\psi\|
$$ exists and is finite. Does it imply that the vector $\psi$ is in the domain of $P$. 
Remark: The expression $\exp(-\epsilon^2 P^2/2) P\psi$ make sense for any $\psi\in\mathcal{H}$ because the operator $\exp(-\epsilon^2 P^2/2) P$ is bounded.
Is the following implication true
$$
\int d (\psi,E_\lambda \psi) ~\lambda^2 < \infty ~~\Rightarrow~~ \psi \in Dom(P)
$$
($E_\lambda$ - spectral family of $P$).
 A: The answer is yes.  This is particularly easy to prove using the "multiplication operator" version of the spectral theorem, for which a good reference is Section VIII.3 of Reed and Simon.
Consider the special case that $\mathcal{H} = L^2(X,\mu)$ for some finite measure space $(X,\mu)$, and that $P$ is multiplication by some real-valued measurable function $h$, i.e. $(Pf)(x) = h(x)f(x)$, where the domain of $P$ is $\{f \in L^2(X,\mu) : hf \in L^2(X,\mu)\}$.  Then $e^{-\epsilon^2 P^2/2} P$ is multiplication by $e^{-\epsilon^2 h^2/2}h$.  Now by Fatou's lemma we have
$$ \int |hf|^2\,d\mu \le \liminf_{\epsilon \to 0} \int \left|e^{-\epsilon^2 h^2/2} hf\right|^2\,d\mu = \liminf_{\epsilon \to 0} \|e^{-\epsilon^2 P^2/2} Pf\|^2$$
which by assumption is finite.  Hence $f$ is in the domain of $P$.
Now the aforementioned spectral theorem states that every self-adjoint operator on a Hilbert space is unitarily equivalent to such a multiplication operator.  That is, given $\mathcal{H}$ and $P$ there is a finite measure space $(X,\mu)$ and a unitary map $U : \mathcal{H} \to L^2(X,\mu)$ such that $UPU^{-1}$ is a multiplication operator of the form described above.  So in fact the result holds in general.
The equation in your final sentence is also true, and again, easiest to check for a multiplication operator.  If $P$ is multiplication by $h$ then it's easy to verify that the spectral family is defined by $E_A$ being multiplication by $1_A(h)$, where $A \subset \mathbb{R}$ is Borel.  It follows that $\int g d(f, E_{\lambda} f) = \int |f|^2 g(h)\,d\mu$, and so when $g(\lambda) = \lambda^2$ we get $\int |fh|^2\,d\mu$, which is finite iff $f$ is in the domain of $P$.
