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A one-dimensional spin-orbit coupled nanowire with proximity-induced pairingfrom a nearby s-wave superconductor may be in a topological nontrivial state,in which it has a zero energy Majorana bound state at each end. We find thatthe topological trivial phase may have fermionic end states with anexponentially small energy, if the confinement potential at the wire's ends issmooth. The possible existence of such near-zero energy levels implies that themere observation of a zero-bias peak in the tunneling conductance is not anexclusive signature of a topological superconducting phase even in the idealclean single channel limit.

Near-zero-energy end states in topologically trivial spin-orbit coupled superconducting nanowires with a smooth confinement