  Based on the quantum two-body problem introduced in [arXiv:1604.06693] we
consider bound pairs of electrons moving on the positive half-line. The
analysis is motivated by the ground-breaking work of Cooper who identified the
pairing of electrons as a possible explanation for superconducting behaviour in
metals. In this paper we are interested in the connection between the
topologies of the underlying one- and two-particle configuration spaces and
spectral properties of the Hamiltonian linked to a condensation of pairs. We
derive explicit estimates for the energy gap and prove condensation for a gas
of non-interacting pairs. Finally, we add some disorder to the system and prove
destruction of the condensate.
