  The purpose of this paper is to introduce the unitary limit as applied in
systems of cold atoms into collective states of heavy, even-even nuclei and to
identify a related physical example. This is accompanied by the determination
of observables of conformal symmetry in nuclear structure. A Hamiltonian is
defined that governs the scattering process of an incident pair of two slow
neutrons onto a heavy, even-even target nucleus in the framework of the
Interacting Boson Model of nuclear structure. A unitary pair-collective state
interaction is introduced along with the effective range expansion of a
pair-collective state scattering length. The solutions to the coupled channels
equations provide a scattering state for the two neutrons that couples with
intermediate states of the $A+2n$ compound nucleus. The unitary limit manifests
itself when the intermediate states correspond to pair-collective state
resonances the positions of which coincide with the energies of IBM states of
the closed channels. The position and the width of each resonance is measurable
via the fluctuation of the cross section which tunes the pair-collective state
scattering length. Conformal symmetry is represented in a tower of equally
spaced states that emerges at the unitary limit from two-boson excitations on
the IBM state of the closed channel. These tower states are experimentally
testable as a regularity pattern of fluctuations of the cross section, in terms
of regular positions and widths. The manifestation of the unitary limit and
conformal symmetry via fluctuations of the cross section indicates the $A+2n$
compound nucleus as a physical laboratory for the examination of the BCS-BEC
crossover, of an underlying critical point and of algebras with infinite number
of generators.
