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\section{Density of states}
In this calculation we reproduce the result for the density of states for a semi-infinite chain of sites under the influence of a spatial dependent potential \cite{Park_2013}. The Hamiltonian for the chain is
\begin{equation}
H_{W}=\sum_{j=1}^{j=N} \epsilon_0 c_{j}^\dagger c_{j} + \sum_{j=N+1}^{j=\Infty} V_g c_{j}^\dagger c_{j} + \sum_{j=1}^{j=\Infty} t\left(c_j^\dagger c_j+1+H.C\right)
\end{equation}
