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.

References in corpus (4)

• On Schroedinger operators with inverse square potentials on the half-line
• 1-dimensional Schrodinger operators with complex potentials
• Bessel-Type Operators and a refinement of Hardy's inequality
• Continuous and holomorphic functions with values in closed operators

Cited by in corpus (6)

• Schrödinger operators with multiple Aharonov-Bohm fluxes
• Bessel-Type Operators and a refinement of Hardy's inequality
• Quantum Geometric Confinement and Dynamical Transmission in Grushin Cylinder
• Aharonov-Casher theorems for Dirac operators on manifolds with boundary and APS boundary condition
• Exactly solvable Schrödinger operators related to the confluent equation
• Heat equation with inverse-square potential of bridging type across two half-lines