Majorana fermions (MFs) are predicted to occur as zero-energy bound states in semiconductor nanowire-superconductor structures. However, in the presence of disorder or smooth confining potentials, these structures can also host non-topological nearly-zero energy states. Here, we demonstrate that the MFs and the nearly-zero topologically-trivial states have different characteristic signatures in a tunneling conductance measurement, which allows to clearly discriminate between them. We also show that low-energy non-topological states can strongly hybridize with metallic states from the leads, which generates the smooth background that characterizes the soft superconducting gap measured in tunneling experiments and produces an additional decoherence mechanism for the Majorana mode. Our results pave the way for the conclusive identification of MFs in a solid state system and provide directions for minimizing quantum decoherence in Majorana wires.
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