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For Riemannian submersions with fibers of basic mean curvature, we comparethe spectrum of the total space with the spectrum of a Schr\"{o}dinger operatoron the base manifold. Exploiting this concept, we study submersions arisingfrom actions of Lie groups. In this context, we extend the state of the artresults on the bottom of the spectrum under Riemannian coverings. As anapplication, we compute the bottom of the spectrum and the Cheeger constant ofconnected, amenable Lie groups.

Spectral estimates for Riemannian submersions with fibers of basic mean curvature