Fate of Majorana zero modes and mobility edges in a one-dimensional quasiperiodic lattice.

We aim to study a one-dimensional $p$-wave superconductor with quasiperiodic on-site potentials. From the open boundary energy spectra, we find there are Majorana zero modes, whose corresponding states are symmetrically distributed at ends of the systems. Further, we numerically calculate the topological invariant by the Pfaffian, confirming that the Majorana zero modes protected by the topology. Moreover, the topological phase transition is accompanied by the energy gap closing. In addition, we numerically find that there are mobility edges, and we qualitatively analyze the influence of superconducting sequence parameters and on-site potential strength on it. In general, our work enriches the study on the $p$-wave superconductor with quasiperiodic potentials.