Back to the periodic case.

From now on, to simplify the exposition, we suppose that
all the gaps of the spectrum of the unperturbed operator are open.
Assumptions on the perturbation:
Pick , define

We assume


So the perturbation is "short range" in the sense that its derivative decays fast enough.

The analytic continuation of the resolvent:
For  and , the resolvent  is a function 
valued in the bounded operators on ; moreover, for any 
it is analytic in a neighborhood of