On the domains of Bessel operators

TL;DR

This paper investigates the domain properties of Bessel operators, a class of Schrödinger operators with inverse square potentials, revealing how the complex parameter m influences their domain structure and Sobolev space relations.

Contribution

It provides a detailed analysis of the domain characterizations of Bessel operators for various complex parameters, clarifying their relation to Sobolev spaces and bilinear forms.

Findings

Domains coincide with minimal Sobolev space for |Re(m)|<1

Unique closed realization domain matches minimal Sobolev space for Re(m)>1

For Re(m)=1, the Sobolev space is of infinite codimension in the domain

Abstract

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.