  We consider a hybrid atom-ion system consisting of a pair of bosons
interacting with a single ion in a quasi-one-dimensional trapping geometry.
Building upon a model potential for the atom-ion interaction developed in
earlier theoretical works, we investigate the behaviour of the low-energy
eigenstates for varying contact interaction strength $g$ among the atoms. In
particular, we contrast the two cases of a static and a mobile ion. Our study
is carried out by means of the Multi-Layer Multi-Configuration Time-Dependent
Hartree method for Bosons, a numerically-exact $\textit{ab initio}$ method for
the efficient simulation of entangled mixtures. We find that repulsive atom
interactions induce locally-distinct modifications of the atomic probability
distribution unique to each eigenstate. Whilst the atoms on average separate
from each other with increasing $g$, they do not necessarily separate from the
ion. The mobility of the ion leads in general to greater separations among the
atoms as well as between the atoms and the ion. Notably, we observe an exchange
between the kinetic energy of the atoms and the atom-ion interaction energy for
all eigenstates, which is both interaction- and mobility-induced. For the
ground state, we provide an intuitive description by constructing an effective
Hamiltonian for each species, which aptly captures the response of the atoms to
the ion's mobility. Furthermore, the effective picture predicts enhanced
localisation of the ion, in agreement with our results from exact numerical
simulations.
