This question follows from this one and especially from Willie Wong's answer: link.
In Reed & Simon's book Methods of modern mathematical physics, vol. I, pag.277, the form domain of a self-adjoint operator $(A, D(A))$ on a Hilbert space $H$ is defined by passing to a spectral representation, that is by taking a unitary isomorphism
$U \colon H \to \bigoplus_{j=1}^N L^2(\mathbb{R}, d\mu_j), $
(where $N\in \{1, 2 \ldots +\infty\}$ and $\mu_j$ are finite Borel measures) such that $(UA)\varphi=(x\psi_j(x))_{j=1}^N$ [$A$ is unitarily equivalent to multiplication by $x$]. The sought domain is then said to be
$Q(q)=\left\{ (\psi_j)_{j=1}^N \ :\ \sum_{j=1}^N \int_{-\infty}^\infty \lvert x \rvert \lvert \psi_j(x)\rvert^2\, d\mu_j <+\infty \right\}.$
Question. Let
$D(\lvert A\rvert^{1/2})=\left\{ \varphi \in H\ :\ \int_{-\infty}^\infty \lvert \lambda \rvert\, d\big(E_A(\lambda)\varphi, \varphi\big)<+\infty\right\},$
where $\{E_A(\lambda)\}_{\lambda \in \mathbb{R}}$ is the spectral family of $A$ (cfr. Reed & Simon vol. I Theorem VIII.6).
Is it true that $Q(q)=D(\lvert A\rvert^{1/2})$?
I believe the answer to be affirmative. This should make for a characterization of the form domain a bit more transparent than the one based on spectral representations. For example, it is not immediately clear that the latter is independent on the particular representation chosen.