# Density of states after scattering

I need some help with a probably simple question because I'm not sure whether my approach is correct.

Let the free Green's function of a system on a discrete lattice described by a massless Dirac Hamiltonian with dispersion $E = v\, p$ be given by

$$G_{0,\vec{p}}^\lambda = \frac{1}{E^2-v^2 p^2 + i\epsilon}\begin{pmatrix} E & \lambda v p e^{i\phi} \\ \lambda v p e^{-i\phi} & E\end{pmatrix},$$

where $E$ is the energy, $v$ the effective speed of the electrons, $p$ the modulus of the 2D momentum vector, $\lambda = \pm1$ the helicity eigenvalue of the electron and $\phi = \arg(p_x+ip_y)$. Let the potential the electrons scatter off be given by

$$V(\vec{p}) = \begin{pmatrix} V_0 & 0 \\ 0 & 0 \end{pmatrix}$$

with $V_0$ constant. The full Green's function up to first order in $V$ is then given by the Dyson equation:

$$G = G_{0,\vec{p}}^\lambda + G_{0,\vec{p}}^\lambda\, V\,G_{0,\vec{p}'}^{\lambda'} = G_{0,\vec{p}}^\lambda + \frac{V_0}{(E^2-v^2p^2+i\epsilon)(E^2-v^2 p'^2+i\epsilon)}\begin{pmatrix} E^2 & E \lambda' v p' e^{i \phi'} \\ E\lambda v p e^{-i\phi} & \lambda\lambda' v^2 p p' e^{i(\phi'-\phi)} \end{pmatrix},$$

where the primed quantities describe the electron after the scattering. In order to get the density of states after one scattering event, I used the relation

$$\rho\left(\vec{p},\vec{p}';E\right) = \frac{1}{\pi} \lim_{\epsilon\to 0}\, \text{Im}\,\text{Tr}\left[ G_{0,\vec{p}}^\lambda\, V\,G_{0,\vec{p}'}^{\lambda'} \right],$$

where $\text{Im Tr}[\ldots]$ denotes the imaginary part of the trace of the argument between the brackets.

First, I would like to know whether this approach is correct. Second, if it is, how do I get rid of the poles in the denominator?