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 |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.
Highlights
We study the domains of various closed homogeneous realizations of the Bessel operator
The Schrodinger operator on the half-line given by the expression α− 1 4
We would like to have a similar construction for Bessel operators, including non-self-adjoint ones
Summary
The theory of closed realizations of (1.1) for complex α is interesting. (Among these realizations, the best known are self-adjoint ones corresponding to real α and real boundary conditions.) All of this is described in [9]. Much more useful are closed realizations of Lα situated between Lmα in and Lmα ax, defined by boundary conditions near zero. Among these realizations for α = 0 only two, and for α = 0 only one, are homogeneous of degree −2.
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