2) for the real and imaginary level shifts as function of the energy difference ∆ i = ϵ i -E (0) . The grey thick line is the unmodified denominator with the singularity at the origin. The purple line is f (∆ i ; ε = 0.2) for the real level shift (the dashed vertical line shows the position of the new singularity). The green line is f (∆ i ; ε = 0.2) for the imaginary level shift.