  Bulk superconductivity (SC) has recently been observed in the Al-Zn-Mg
quasicrystal (QC). To settle several fundamental issues of the SC on the QC, we
use an attractive Hubbard model to perform a systematic study on the Penrose
lattice. The first issue is the Cooper instability of the QC, i.e., no Fermi
surface under an infinitesimal attractive interaction. Starting from the
two-electron problem outside the filled Fermi-sea, we analytically prove that
an infinitesimal Hubbard attraction can lead to the Cooper instability as long
as the density of state is nonzero at the Fermi level, which provides the basis
of the SC on the QC. Our numerical results yield that the Cooper pairing always
takes place between the two time-reversal states, satisfying the Anderson's
theorem. On this theorem, we perform a mean-field (MF) study at both zero and
finite temperatures. The MF study also shows that an arbitrarily weak
attraction can lead to the pairing order, with the resulted pairing state being
well described by the BCS theory and the thermal dynamic behaviors being well
consistent with experimental results. The second issue is about the superfluid
density on the QC without translational symmetry. It's clarified that although
the normal state of the system locates at the critical point of the
metal-insulator transition, the pairing state exhibits real SC, carrying finite
superfluid density that can be verified by the Meissner effect, consistent with
experiment also. These revealed properties of the SC on the Penrose lattice are
universal for all QCs.
