Abstract
AbstractWe consider the Schrödinger operator on the halfline with the potential $$(m^2-\frac{1}{4})\frac{1}{x^2}$$
(
m
2
-
1
4
)
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 $$|\mathrm{Re}(m)|<1$$
|
Re
(
m
)
|
<
1
and of its unique closed realization for $$\mathrm{Re}(m)>1$$
Re
(
m
)
>
1
coincide with the minimal second-order Sobolev space. On the other hand, if $$\mathrm{Re}(m)=1$$
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.
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Nuclear and High Energy Physics,Statistical and Nonlinear Physics
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