  We study the bound states of a three-dimensional Holstein polaron near
various kinds of single impurities, using the momentum average approximation.
We show that the electron-phonon coupling is responsible for a strong
renormalization of the impurity potential, resulting in an effective potential
with significant retardation effects, which describes essential physics ignored
by "instantaneous" approximations. The accuracy of our approximation is gauged
by comparison with results from Diagrammatic Monte Carlo for the case of an
impurity that modifies the on-site energy of the electron. We also discuss
impurities that modify the local strength of the electron-phonon coupling, as
well as isotope substitutions that change both the electron-phonon coupling and
the phonon frequency, and contrast and highlight the difference between these
cases.
