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We investigate the existence of end points in the dispersion of Holsteinpolarons in various dimensions, using the Momentum Average approximation whichhas proved to be very accurate for this model. An end point separates momentafor which the lowest-energy state is a discrete level, i.e., aninfinitely-lived polaron, from those where the lowest-energy feature is acontinuum in which the "polaron'" is signalled by a resonance with a finitelifetime. While such end points are known to not appear in 1D, we show herethat 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 the1D 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 modelsare also briefly discussed.

The end points in the dispersion of Holstein polarons