Upper and lower bounds for the large polaron dispersion in $D=1,2,3$ dimensions

Numerical results for the polaron dispersion are presented for an arbitrary number of space dimensions. Upper and lower bounds are calculated for the dispersion curves. They are rather close to each other in the cases of small electron-phonon couplings usual for real polar materials. To describe the dispersion in other materials, we suggest a simple fitting formula which can be applied at intermediate values of the Fr"ohlich electron-phonon coupling constant. Its validity is approved by the comparison with direct calculations and previously obtained results. This makes our results not only reliable and highly accurate but also easy reproducible.