On the LO-polaron dispersion in D dimensions

We discuss the (LO)polaron dispersion for arbitrary spatial dimension D. Firstly, we review the existing literature; recent numerical work is critically analyzed. Secondly, we derive novel upper bounds for the dispersion, which incorporate the correct behaviour of the dispersion up to third order of the coupling constant a. A totally analytical evaluation is performed in the case D = 1. We compare the upper bounds with previously published lower bounds. Apart from a surrounding of zero dispersion, the relative deviation is on a few-percent scale.