Operateurs de Schrodinger quasi-periodiques adiabatiques : Interactions   entre les bandes spectrales d'un operateur periodique
  This paper is devoted to the description of our recent results on the spectral behavior of one-dimensional adiabatic quasi-periodic Schrodinger operators. The specific operator we study is a slow periodic perturbation of an incommensurate periodic Schrodinger operator, and we are interested in energies where the perturbation creates a strong interaction between two consecutive bands of the background periodic operator. We describe the location of the spectrum and its nature and discuss the various new resonance phenomena due to the interaction of the spectral bands of the unperturbed periodic operator. 
Alexandre Fedotov, Frederic Klopp