  Non-local, inhomogeneous and retarded response observed in experiments is
reproduced by introducing the Inhomogeneous Momentum Average (IMA) method to
study single polaron problems with disorder in the on-site potential and/or
spatial variations of the electron-phonon couplings and/or phonon frequencies.
We show that the electron-phonon coupling gives rise to an additional
inhomogeneous, strongly retarded potential, which makes instant approximations
questionable. The accuracy of IMA is demonstrated by comparison with results
from the approximation free Diagrammatic Monte Carlo (DMC) method. Its
simplicity allows for easy study of many problems that were previously
unaccessible. As an example, we show how inhomogeneities in the electron-phonon
coupling lead to nonlocal, retarded response in scanning tunneling microscopy
(STM) images.
