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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=\Infinity} \sum_{j=N+1}^{j=\infty}  V_g c_{j}^\dagger c_{j} + \sum_{j=1}^{j=\Infinity} \sum_{j=1}^{j=\infty}  t \left(c_j^\dagger c_j+1+H.C\right) \end{equation}