On the Domains of Bessel Operators

We consider the Schrodinger operator on the halfline with the potential $$(m^2-\frac{1}{4})\frac{1}{x^2}$$ , 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 $$|\mathrm{Re}(m)|<1$$ and of its unique closed realization for $$\mathrm{Re}(m)>1$$ coincide with the minimal second-order Sobolev space. On the other hand, if $$\mathrm{Re}(m)=1$$ 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.