Consider

where
 is a real parameter;
 and  are non constant, periodic real valued functions i.e.

 is locally square integrable and  is analytic in a neighborhood of the real axis.

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

Adiabatic equations when  is small: lots of works too (Buslaev, Dimassi, Dimitrieva,
Gérard, Grigis, Helffer, Poupaud, Sjöstrand, Papanicolaou, Spohn,...).