I have a small bit of confusion about the expansion I am seeing in literature for the Greens function in time independent scattering theory. For example here is an excerpt from Scattering Theory of Waves and Particles by R.G. Newton:
My question is quite simple. Why in equation 7.26 is there no $\pm i\epsilon$ in the denominator in the first term, i.e. the discrete sum over bound states. I understand that for most energies the $\epsilon$ in this term may be simply ignored, but what if $E$ is exactly equal to some $E_n$, then we have a pole at this energy right? So does this equation have some implicit assumption that we do not take $E$ exactly at the bound state energy? Or is there some other reason?