Computing the resonances:

Fix an energy, say  satisfying the assumptions of Theorem 1.

It is well known that the existence and the positions of resonances will depend crucial-
ly on the relative position of the spectral window  with respect to 
 (see the work of Dimassi).
Introduce the set 

As  satisfies the assumptions of Theorem 1, a simple Weyl sequence argument then
shows that  in the interior of the spectrum of the total operator.

Two types of situations and two different assumptions