  We study the low-energy physics of a one-dimensional array of superconducting
quantum dots realized by proximity coupling a semiconductor nanowire to
multiple superconducting islands separated by narrow uncovered regions. The
effective electrostatic potential inside the quantum dots and the uncovered
regions can be controlled using potential gates. By performing detailed
numerical calculations based on effective tightbinding models, we find that
multiple low-energy sub-gap states consisting of partially overlapping Majorana
bound states emerge generically in the vicinity of the uncovered regions.
Explicit differential conductance calculations show that a robust zero-bias
conductance peak is not inconsistent with the presence of such states localized
throughout the system, hence the observation of such a peak does not
demonstrate the realization of well-separated Majorana zero modes. However, we
find that creating effective potential wells in the uncovered regions traps
pairs of nearby partially overlapping Majorana bound states, which become less
separated and acquire a finite gap that protects the pair of Majorana zero
modes localized at the ends of the system. This behavior persists over a
significant parameter range, suggesting that proximitized quantum dot arrays
could provide a platform for highly controllable Majorana devices.
