  We investigate a temporal evolution of an impurity atom in a one-dimensional
trapped Bose gas following a sudden change of the boson-impurity interaction
strength. Our focus is on the effects of inhomogeneity due to the harmonic
confinement. These effects can be described by an effective one-body model
where both the mass and the spring constant are renormalized. This is in
contrast to the classic renormalization, which addresses only the mass. We
propose an effective single-particle Hamiltonian and apply the multilayer
multiconfiguration time-dependent Hartree method for bosons to explore its
validity. Numerical results suggest that the effective mass is smaller than the
impurity mass, which means that it cannot straightforwardly be extracted from
translationally invariant models.
