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 i.e.


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,...)

We are interested in the adiabatic case i.e. the case when  is small.
We assume  is analytic.