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We investigate the interplay between disorder and superconducting pairing fora one-dimensional $p$-wave superconductor subject to slowly varyingincommensurate potentials with mobility edges. With amplitude increments of theincommensurate potentials, the system can undergo a transition from atopological phase to a topologically trivial localized phase. Interestingly, wefind that there are four mobility edges in the spectrum when the strength ofthe incommensurate potential is below a critical threshold, and a noveltopologically nontrivial localized phase emerges in a certain region. We revealthis energy-dependent metal-insulator transition by applying several numericaldiagnostic techniques, including the inverse participation ratio, the densityof states and the Lyapunov exponent. Nowadays, precise control of thebackground potential and the $p$-wave superfluid can be realized in theultracold atomic systems, we believe that these novel mobility edges can beobserved experimentally.

Fate of topological states and mobility edges in one-dimensional slowly varying incommensurate potentials