  Majorana bound states (MBS) and Andreev bound states (ABS) in realistic
Majorana nanowires setups have similar experimental signatures which make them
hard to distinguishing one from the other. Here, we characterize the continuous
Majorana/Andreev crossover interpolating between fully-separated,
partially-separated, and fully-overlapping Majorana modes, in terms of global
and local topological invariants, fermion parity, quasiparticle densities,
Majorana pseudospin and spin polarizations, density overlaps and transition
probabilities between opposite Majorana components. We found that in
inhomogeneous wires, the transition between fully-overlapping trivial ABS and
nontrivial MBS does not necessarily mandate the closing of the bulk gap of
quasiparticle excitations, but a simple parity crossing of partially-separated
Majorana modes (ps-MM) from trivial to nontrivial regimes. We demonstrate that
fully-separated and fully-overlapping Majorana modes correspond to the two
limiting cases at the opposite sides of a continuous crossover: the only
distinction between the two can be obtained by estimating the degree of
separations of the Majorana components. This result does not contradict the
bulk-edge correspondence: Indeed, the field inhomogeneities driving the
Majorana/Andreev crossover have a length scale comparable with the nanowire
length, and therefore correspond to a nonlocal perturbation which breaks the
topological protection of the MBS.
