Magnetic Schr\"odinger operators and Ma\~n\'e's critical value

We study periodic magnetic Schr\"odinger operators on covers of closed manifolds in relation to Ma\~n\'e's critical energy values of the corresponding classical Hamiltonian systems. In particular, we show that if the covering transformation group is amenable, then the bottom of the spectrum is bounded from above by Ma\~n\'e's critical energy value. We also determine the spectra for various homogeneous spaces with left-invariant magnetic fields.