  We investigate the existence of end points in the dispersion of Holstein
polarons in various dimensions, using the Momentum Average approximation which
has proved to be very accurate for this model. An end point separates momenta
for which the lowest-energy state is a discrete level, i.e., an
infinitely-lived polaron, from those where the lowest-energy feature is a
continuum in which the "polaron'" is signalled by a resonance with a finite
lifetime. While such end points are known to not appear in 1D, we show here
that they are generic in 3D if the particle-boson coupling is not too strong.
The 2D case is "critical": a pure 2D Holstein model has no end points, like the
1D case. However, any amount of interlayer hopping leads to 3D-like behavior.
As a result, such end points are expected to appear in the spectra of layered,
quasi-2D systems described by Holstein models. Generalizations to other models
are also briefly discussed.

Strongly Correlated Electrons