Dynamic correlation functions for integrable mobile impurities

in terms of the corresponding Bethe Ansatz solution and demonstrate, in the leading order, the absence of two-phonon scattering resulting in vanishing rates of inelastic processes such as viscosity experienced by the mobile impurities

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N

and the superfluid phase drop

\pi J

c=1

. For higher energy excitations the spectral function displays deviations from the Luttinger behavior arising from the curvature terms in the dispersion. Adding a corrective term of the form of a mobile impurity coupled to the Luttinger liquid modes corrects this problem. The ``impurity'' term is an irrelevant operator, which if treated non-perturbatively, yields the threshold singularities in the one-particle and one-hole hole Green's function correctly. We show that the exponents obtained via the finite size corrections to the ground state energy are identical to those obtained through the shift function.