and (e-f) enhanced ($\delta \Delta=+0.05t$) superconducting gap. The Ni potentials used for this study are $V_3=-0.8t$, $V_0=0.6t$. The figures in the left column (a,c,e) show spectra on the impurity site (IS) itself, while right column (b,d,f) represents nearest neighbor (NN) sites. In each case, the solid black curve shows the LDOS on the site in question in the absence of the impurity, the solid color curve indicates the total LDOS with the impurity, and the dashed and dotted lines show the spin resolved LDOS for up and down spins, respectively.[]{data-label="fig1"}](Fig1.eps){width="1.0\columnwidth"}

To obtain the resulting LDOS as a function of position and energy, one needs to determine the full Green’s function ${\mathcal{G}}({\mathbf{r}},i\omega_n)$ given by the Dyson equation $$\label{dyson}
{\mathcal{G}}({\mathbf{r}},{\mathbf{r}}')={\mathcal{G}}^{0}({\mathbf{r}}-{\mathbf{r}}')+
{\mathcal{G}}({\mathbf{r}},{\mathbf{r}}'') {\cal
H}^{\prime}({\mathbf{r}}'',{\mathbf{r}}''')
{\mathcal{G}}^{0}({\mathbf{r}}'''-{\mathbf{r}}'),$$ where ${\cal H}^\prime ={\cal H}^\prime_{imp}+{\cal
H}^\prime_\delta$, and summation over repeated indices is implied. Thus, by calculating the matrix elements of ${\mathcal{G}}^{0}({\mathbf{r}},i\omega_n)=
\sum_{\mathbf{k}}\mathcal{G}^{0}({\mathbf{k}},i\omega_n)e^{i
{\mathbf{k}} \cdot {\mathbf{r}}}$ the remaining problem is that of a matrix inversion. The solution is presented in terms of the [$\cal T$]{}-matrix $$\label{tmat}
{\mathcal{G}}({\mathbf{r}},{\mathbf{r}}')={\mathcal{G}}^{0}({\mathbf{r}}-{\mathbf{r}}')+
{\mathcal{G}}^{0}({\mathbf{r}}-{\mathbf{r}}'') {\cal
T}({\mathbf{r}}'',{\mathbf{r}}''')
{\mathcal{G}}^{0}({\mathbf{r}}'''-{\mathbf{r}}').$$ The poles of the ${\cal T}$-matrix, or equivalently, the zeros in the determinant of $(1-{\cal H}^{\prime}{\cal G}^{0})$, determine the bound state energies. The total (spin summed) LDOS is 