Schrodinger spectra and the effective Hamiltonian of weak KAM theory on the flat torus

In this paper we investigate the link between the spectrum of some periodic Schrodinger type operators and the effective Hamiltonian of the weak KAM theory. We show that the extension of some local quasimodes is linked to the localization of the Schrodinger spectrum. Such a result provides additional information with respect to the well known Bohr-Sommerfeld quantization rules, here in a more general setting than the integrable or quasi-integrable ones. Published by AIP Publishing.