  We consider the Schr\"odinger operator on the halfline with the potential
$(m^2-\frac14)\frac1{x^2}$, often called the Bessel operator. We assume that
$m$ is complex. We study the domains of various closed homogeneous realizations
of the Bessel operator. In particular, we prove that the domain of its minimal
realization for $|\Re(m)|<1$ and of its unique closed realization for
$\Re(m)>1$ coincide with the minimal second order Sobolev space. On the other
hand, if $\Re(m)=1$ the minimal second order Sobolev space is a subspace of
infinite codimension of the domain of the unique closed Bessel operator. The
properties of Bessel operators are compared with the properties of the
corresponding bilinear forms.
