Consider the case of a relativistic electron on a graphene lattice described by the Hamiltonian

$$ \mathcal{H} = v\begin{pmatrix} 0 & p_x+ip_y \\ p_x-ip_y & 0 \end{pmatrix}, $$

where $v$ is the effective velocity of the electron and $p_x$, $p_y$ are the components of the 2D momentum vector $\vec{p}$. This Hamiltonian is valid for $|\vec{p}| \ll 1$. Diagonalization of this matrix gives us the eigenvalues $E = \pm v|\vec{p}|$ and the eigenvectors

$$ \Psi_0(\vec{p}) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ \pm e^{i\phi_\vec{p}} \end{pmatrix} e^{-\frac{i}{\hbar}\vec{p}\cdot\vec{r}} $$

with $\phi_p = \tan^{-1}(p_y/p_x)$.

I now want to use scattering theory to determine the scattering amplitude of an electron scattering off a potential of the form $V(\vec{r}) = V_0\delta(\vec{r})$ with the Dirac delta distribution $\delta(\vec{r})$ and $V_0 > 0$. My plan was to use the Lippmann-Schwinger equation which in first order relates the total wave function after the scattering event $\Psi$ to the incident wave function $\Psi_0$ and the scattered wave function $\Psi_1$ by

$$ \Psi = \Psi_0 + G_0V\Psi_0. $$

$G_0$ is the free Green's function defined by $(\mathcal{H}-E)G_0=1$. So $G_0 = (\mathcal{H}-E)^{-1}$, which is a problem, because

$$ \mathcal{H}-E = v\begin{pmatrix} -|\vec{p}| & p_x+ip_y \\ p_x-ip_y & -|\vec{p}| \end{pmatrix} $$

is singular, choosing $E = +v|\vec{p}|$. So, there must be something wrong in my approach to the problem. This is the first issue.

Let's assume, we can find some $G_0$. How do I then implement the scattering potential $V(\vec{r})$ in order to get the scattering amplitude with an angular dependence?

I hope there will be someone who can clarify the two issues.



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