On the domains of Bessel operators

Abstract

We consider the Schrödinger 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 $|\Re (m)|<1$ 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.