On the Domains of Bessel Operators

Abstract

We consider the Schrödinger operator on the halfline with the potential $$(m^2-\frac{1}{4})\frac{1}{x^2}$$ ( 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.

Self-adjoint extension schemes and modern applications to quantum Hamiltonians

“…Concerning A α (0), it suffices to import from the literature the following analogue of Lemma 5.11. Proof A direct consequence of [77,Theorem 4.1]: in the notation therein A α (0) is the operator L min δ with δ − 1 4 = C α (the present δ replaces the notation α from [77] so as not to clash with the current meaning of α), that is δ = ( 1+α 2 ) 2 ; the requirement Re √ δ < 1 needed for the applicability of [77, Theorem 4.1] is therefore satisfied, since α ∈ (0, 1).…”

“…The characterisation of the distinguished extension of h declared in Proposition 4.3 goes through the differential problem h f = g in the unknown f for given g ∈ L 2 (R + , C), so as to determine the domain of invertibility of the differential operator h. The strategy here is an adaptation to the present first order differential operator with Coulomb singularity of the analogous problem for homogeneous Schrödinger operators of Bessel type on half-line, a subject that is well classical [51,77] and which is also encountered, in a different context, in Sections 3.2.1, 5.6 and 5.8.6.…”

“…therefore, when |ξ | < 1 one has |W ξ ,−1 (x)| < 3 4x 2 for all x > 0, thus A + α (ξ ) is in the limit-circle case at zero and hence it has unit deficiency index, whereas when instead |ξ | 1 one has W ξ ,−1 (x) 3 4x 2 for all x > 0, thus A + α (ξ ) is in the limit-point case at zero and hence it is essentially self-adjoint; in fact, A −1 (ξ ) is precisely the minimally defined Bessel operator, for which it is well-known (see, e.g., [77]) that it is essentially self-adjoint for |ξ | 1 and has unit defi-…”

“…In fact, the zero mode fibre operator A α (0) is a classical Bessel operator and its self-adjoint realisations are known in the literature, obtained by other means and from a different perspective (see, e.g., [51,77]): an amount of details will be therefore omitted here.…”

Bessel-Type Operators and a refinement of Hardy's inequality

“…implying S * 0,min = S 0,max , S * 0,max = S 0,min , and we also introduce the following self-adjoint extensions of S 0,min , respectively, restrictions of S 0,max (see, e.g., [3], [5], [4], [11], [15], [19], [24], [28], [38], [40], [56]),…”