Consider the family of equations



where
 is a real parameter indexing the equations of the family;
 and  are real-valued, locally , non constant, periodic functions



Motivation: 
                     modeling single particle in a disorder medium.

Family of quasi-periodic equations on :  during the last 30 years, huge amount of 
work on their spectral theory (Avron, Bellissard, Bourgain, Buslaev, Chulaevsky,
Damanik, Dinaburg,  Eliasson, Helffer, Hermann, Jitomirskaya, Last, Shubin, Simon, 
Sinaï, Sjöstrand, Sorets, Spencer, Sütö, Wilkinson,...)

Pb: understand the spectral properties.