  For Riemannian submersions with fibers of basic mean curvature, we compare
the spectrum of the total space with the spectrum of a Schr\"{o}dinger operator
on the base manifold. Exploiting this concept, we study submersions arising
from actions of Lie groups. In this context, we extend the state of the art
results on the bottom of the spectrum under Riemannian coverings. As an
application, we compute the bottom of the spectrum and the Cheeger constant of
connected, amenable Lie groups.
