The behavior of the real part of the resonances in 
It is the behavior of resonances when one shifts the perturbation along the periodic 
potential:

In Marx's thesis, she showed that the eigenvalues created in the gaps of the periodic 
operator oscillate with an exponentially small amplitude. 

For resonances, it depends of the value of 
if  then the oscillations are again exponentially small.
if  then the oscillations are of order one.