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

TL;DR

This paper investigates the spectral properties of periodic magnetic Schr"odinger operators on manifold covers, linking the spectrum's lower bound to Ma\'n\'e's critical energy, especially under amenable group actions.

Contribution

It establishes a connection between the spectrum's bottom and Ma\'n\'e's critical value for amenable covers and characterizes spectra on homogeneous spaces with invariant magnetic fields.

Findings

Spectrum bottom is bounded by Ma\'n\'e's critical value for amenable groups.

Spectra are explicitly determined for certain homogeneous spaces.

The results bridge classical Hamiltonian dynamics and quantum spectral theory.

Abstract

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.