A few words on resonances.

One wants to construct an meromorphic continuation of the resolvent.
Useful to get information on the time asymptotics


With analytic continuation of the resolvent, one deforms 
the contour



In some cases, for large , the last integral can be estimated by terms of order 
So if the imaginary part of the resonances is small, for some time, the behavior of the 
semigroup is given by the first terms in the sum.
Resonances have been studied a lot:
for local perturbations of the Laplacian: Aguilar, Balslev, Burq, Combes, Gérard,
Martinez, Melrose, Petkov, Simon, Sjöstrand, Soffer, Stefanov, Vodev, Zworski, ...
for local perturbations of periodic operators: Buslaev, Dimassi, Firsova, Gérard, 
Grigis, K., Korotyaev, Nier, Zerzeri, ...