On the Domains of Bessel Operators

We consider the Schrödinger operator on the halfline with the potential (m2−14)1x2, 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.