On the domains of Bessel operators

Abstract

We consider the Schrödinger operator on the halfline with the potential (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 |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.

On domain properties of Bessel-type operators

“…In this context we note that (4.10) can be found in [8], [12] (for the interval (0, ∞)), [25], [33], [36] (also for the interval (0, ∞)), for (4.11) we refer to [16], [25], [33], for (4.12) to [19], [33], [41], [46], and in connection with (4.13) to [4], [8], [12], [25], [33]. Relations (4.10)-(4.14) do by no means exhaust the possible descriptions of S sa,F and dom S 1/2 sa,F and more can be found, for instance, in [3], [5], [4], [8], [12], [16], [26], [33], [36], [46].…”

“…We emphasize that more can and has been proven in this context in [8] (see also [3], [4], [5], [12], [36]) using quite different methods, not involving Hardy-type inequalities such as (B.2). In particular, the case of complex-valued s a is considered in [8], [12]. Remark 3.3.…”

“…Closest to the results presented in this paper is the work by A. Yu. Annan'eva and V. S. Budyka [3], [4], [5], and by L. Bruneau, J. Derezinski, and V. Georgescu [8], [12], which focus on Bessel operators on (0, ∞) and (0, b), b ∈ (0, ∞), with inverse square singularity at x = 0. The techniques used by these authors are based on elements of harmonic analysis and the explicit knowledge of resolvents for firstorder differential operators, respectively, while our methods predominantly rely on weighted Hardy inequalities.…”

Closure of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds and semi-Fredholm properties of differential operators on Lie manifolds

“…Operators similar to (12) are often encountered in practice and were extensively studied in the past. Let us compare Propositions 6.1 and 6.2 with results which exist in the literature, more precisely with [28] which has a literature overview and the most up-todate results. In [28] the authors study the operator of the form…”

“…If α = 0, then both ∆ and L 1 are self-adjoint real symmetric operators. One can check that functions that go to zero as O(x 3/2 ) when x → 0+ lie in the domain of the adjoint (those are L 2 functions mapped to L 2 functions) and hence lie in the domain of the closure as well by the results of [28]. This asymptotics is indeed consistent with Proposition 6.2 and is also in accordance with the results of [37], which can be seen as the generalisation of operators (12) and (14).…”