Consider the following family of Schrödinger operators acting on 

where
is a non constant, locally square integrable, 1-periodic function;
 is a small positive number;
 is a real parameter;
 is a potential that is real analytic in a conic neighborhood of the real axis 
and that admits limits at both ends of the real axis.
The spectrum of the periodic operator

is purely absolutely continuous and made of bands.


Our goal: 
compute the width of the resonances sitting near the bands of the unperturbed operator.