Non-gapless excitation and zero-bias fast oscillations in the LDOS of surface superconducting states

Recently a novel surface pair-density-wave (PDW) superconducting state has been discovered in Refs. [Phys. Rev. Lett. \textbf{122}, 165302 (2019)] and Phys. B \textbf{101}, 054506 (2020)], which may go through distinct multiple phase transition (MPT) when the superconductivity fades away from bulk to boundary (e.g. edges corners). Based on Bogoliubov-de Gennes equations for attractive tight-binding Hubbard modal one-dimensional chain, we demonstrate that PDW non-gapless quasiparticle spectrum, is contrary conventional state. Moreover, find MPT associated with zero-bias fast oscillating pattern LDOS near surface. Our findings provide potential experimental clue identify