  We present a comprehensive study of the static properties of a mobile
impurity interacting with a bath with a few particles trapped in a
one-dimensional harmonic trap. We consider baths with either identical bosons
or distinguishable particles and we focus on the limiting case where the bath
is non-interacting. We provide numerical results for the energy spectra and
density profiles by means of the exact diagonalization of the Hamiltonian, and
find that these systems show non-trivial solutions, even in the limit of
infinite repulsion. A detailed physical interpretation is provided for the
lowest energy states. In particular, we find a seemingly universal transition
from the impurity being localized in the center of the trap to being expelled
outside the majority cloud. We also develop an analytical ansatz and a
mean-field solution to compare them with our numerical results in limiting
configurations.
