to a patch of modulated $\Delta_\k$ of roughly 20 Å. Thus we study the impurity added to a patch of reduced or enhanced $\Delta_\k$ on a 5$\times$5 square as shown in Fig. \[schematic\] embedded in an infinite system of fixed order parameter $\Delta_0$.

N impurity
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We first discuss the case of magnetic impurity ions such as Ni. In Fig. \[fig1\] we show the LDOS at the magnetic impurity site using the parameters $V_3=-0.8t$,$V_0=0.6t$, and $\delta
\Delta=-0.05t, 0, +0.05t$ corresponding to a local suppressed, constant, and enhanced gap patch, respectively. The ${\cal T}$-matrix corresponding to this potential has a pole at an energy $\Omega_0\sim0.05t$, but, as is evident from Fig. \[fig1\], resonance are visible neither on the impurity site (IS) nor on the nearest neighbor (NN) site. Note that while in Fig. \[fig1\] we have assumed a quasiparticle scattering rate $\Gamma(\omega)=0.1|\omega|$ similar to that observed in experiment[@alldredge], the absence of resonance features is not caused by this broadening. Instead, it is the fact that the coupling to the continuum at $\Omega_0$ is simply too strong. In Fig. \[fig2\], we show the same sites with the same potentials in a situation where the hopping has been reduced around the Ni impurity; 