  We investigate the ground-state energy of a one-dimensional Fermi gas with
two bosonic impurities. We consider spinless fermions with no fermion-fermion
interactions. The fermion-impurity and impurity-impurity interactions are
modelled with Dirac delta functions. First, we study the case where impurity
and fermions have equal masses, and the impurity-impurity two-body interaction
is identical to the fermion-impurity interaction, such that the system is
solvable with the Bethe ansatz. For attractive interactions, we find that the
energy of the impurity-impurity subsystem is below the energy of the bound
state that exists without the Fermi gas. We interpret this as a manifestation
of attractive boson-boson interactions induced by the fermionic medium, and
refer to the impurity-impurity subsystem as an in-medium bound state. For
repulsive interactions, we find no in-medium bound states. Second, we construct
an effective model to describe these interactions, and compare its predictions
to the exact solution. We use this effective model to study non-integrable
systems with unequal masses and/or potentials. We discuss parameter regimes for
which impurity-impurity attraction induced by the Fermi gas can lead to the
formation of in-medium bound states made of bosons that repel each other in the
absence of the Fermi gas.
