  The effect of a Coulombic coupling on the dynamics of a quantum dot
hybridized to leads is determined. The calculation treats the interaction
between charge fluctuations on the dot and the dynamically generated image
charge in the leads. A formally exact solution is presented for a dot coupled
to a Luttinger liquid and an approximate solution, equivalent to treating the
lead dynamics within a random phase approximation, is given for a dot coupled
to a two- or three-dimensional metallic lead. The leading divergences arising
from the long-ranged Coulomb interaction are found to cancel, so that in the
two- and three-dimensional cases the quantum-dot dynamics is equivalent to that
obtained by neglecting both the dot-lead Coulomb coupling and the Coulomb
renormalization of the lead electrons, while in the one-dimensional case the
dot-lead mixing is enhanced relative to the non-interacting case. Explicit
results are given for the short-time dynamics.
