  Background: The pairing correlation in weakly bound nuclei causes a mixing
among bound and unbound configurations. A remarkable consequence is emergence
of the quasiparticle resonance, which has been predicted with the coordinate
space Hartree-Fock-Bogoliubov (HFB) theory, but not yet observed
experimentally. Purpose: We discuss possible observation of quasiparticle
resonances in decay spectrum of unbound nuclei near the neutron drip-line. We
deal with an example of unbound nucleus ${}^{21}$C which disintegrates to
${}^{20}$C and a neutron. Method: We describe a scattering state consisting of
${}^{20}$C and a neutron in the framework of the HFB formalism. We assume that
a nucleon knockout reaction produces a doorway state of the decay, and we
evaluate the decay spectrum by taking an overlap of the doorway state and the
scattering state of ${}^{20}\mathrm{C}+n$. A numerical calculation was
performed with the Woods-Saxon potential and a density-dependent effective
pairing interaction. Results: We show that the quasiparticle resonance appears
as low-lying peaks in the decay spectrum of ${}^{21}$C. They originate from the
weakly bound single-neutron orbits $2s_{1/2}$ and $1d_{5/2}$, but emerge as
unbound resonant quasiparticle states under the influence of the neutron
pairing correlation. The resonance energy and the width of the calculated
quasiparticle resonances are consistent with an experimental observation
whereas they are sensitive to the neutron pairing correlation. Conclusion: The
results suggest that nucleon knockout reactions populating the unbound nucleus
${}^{21}\mathrm{C}$ provide realistic opportunity of experimentally observing
the quasiparticle resonance and of disclosing the pairing correlation in
neutron-rich nuclei.
