Title: Coalescence of Two Impurities in a Trapped One-dimensional Bose Gas,
Abstract: We study the ground state of a one-dimensional (1D) trapped Bose gas with two
mobile impurity particles. To investigate this set-up, we develop a variational
procedure in which the coordinates of the impurity particles are slow-like
variables. We validate our method using the exact results obtained for small
systems. Then, we discuss energies and pair densities for systems that contain
of the order of one hundred atoms. We show that bosonic non-interacting
impurities cluster. To explain this clustering, we calculate and discuss
induced impurity-impurity potentials in a harmonic trap. Further, we compute
the force between static impurities in a ring ({\it {à} la} the Casimir
force), and contrast the two effective potentials: the one obtained from the
mean-field approximation, and the one due to the one-phonon exchange. Our
formalism and findings are important for understanding (beyond the polaron
model) the physics of modern 1D cold-atom systems with more than one impurity.