  The problem of the Kohn mode in bosonized theories of one-dimensional
interacting fermions in the harmonic trap is investigated and a suitable
modification of the interaction is proposed which preserves the Kohn mode. The
modified theory is used to calculate exactly the inhomogeneous linear mobility
at position z in response to a spatial force pulse at another position. It is
found the inhomogeneous particle mobility exhibits resonances not only at the
trap frequency but also at multiples of a new renormalized collective mode
frequency which depends on the strength of the interaction. In contrast, the
local response obtained by averaging over the pulse position remains that of
the non-interacting system.
