  In a recent work, the authors studied various Borel equivalence relations
defined on the Polish space ${\rm{SA}}(H)$ of all (not necessarily bounded)
self-adjoint operators on a separable infinite-dimensional Hilbert space $H$.
In this paper we study the domain equivalence relation
$E_{\rm{dom}}^{{\rm{SA}}(H)}$ given by
$AE_{\rm{dom}}^{{\rm{SA}}(H)}B\Leftrightarrow {\rm{dom}}{A}={\rm{dom}}{B}$ and
determine its exact Borel complexity: $E_{\rm{dom}}^{{\rm{SA}}(H)}$ is an
$F_{\sigma}$ (but not $K_{\sigma}$) equivalence relation which is continuously
bireducible with the orbit equivalence relation
$E_{\ell^{\infty}}^{\mathbb{R}^{\mathbb{N}}}$ of the standard Borel group
$\ell^{\infty}=\ell^{\infty}(\mathbb{N},\mathbb{R})$ on
$\mathbb{R}^{\mathbb{N}}$. This, by Rosendal's Theorem, shows that
$E_{\rm{dom}}^{{\rm{SA}}(H)}$ is universal for $K_{\sigma}$ equivalence
relations. Moreover, we show that generic self-adjoint operators have purely
singular continuous spectrum equal to $\mathbb{R}$.
