Bulk SC has recently been observed in the Al-Zn-Mg QC. To settle the several fundamental issues on the SC on the QC, we perform a systematic study on an attractive Hubbard model on the Penrose lattice. The first issue is the Cooper instability under an infinitesimal attractive interaction on the QC without a Fermi surface. We start from the two-electron problem outside the filled Fermi-sea, where 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 for the SC on the QC. Our numerical results yield that the Cooper pairing always takes place between a time-reversal partner, satisfying the Andersons theorem. On this basis, we perform a MF study on the system, at both the zero and finite temperatures. The MF study also shows that an arbitrarily weak attraction can lead to the pairing order, with the resulting pairing state well described by the BCS theory, and the thermal dynamic behaviors well consistent with experiment results. The second issue is about the superfluid density on the QC without translational symmetry. Its 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. Further more, our study reveals a fundamental difference between the SC on the periodic lattice and that on the QC: while the paramagnetic superfluid density in the former case vanishes at zero temperature, that in the latter case is nonzero due to the lack of translational symmetry, reflecting the consumption of superfluid density from the scattering by the non-periodic structure. These properties of the SC on the Penrose lattice revealed here are universal for all QCs.
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