that the images at positive and negative bias show fourfold patterns which are rotated by 45$^\circ$ with respect to each other, as observed in experiment[@lang] and earlier theories.[@tangflatte; @balatsky]

![LDOS real-space pattern $N({\bf r},\pm \Omega_0)$ at the resonance frequency for Ni impurities modelled as described in the text. Left panel: no suppression of hopping by impurity assumed. Subpanels (a-b), (c-d), and (e-f) show the resonance pattern for gap sizes in the surrounding 5$\times$5 patch corresponding to $\delta \Delta=+0.05t,0,\mbox{and} -0.05t$, respectively. Left column (a,c,e) and right column (b,d,f) correspond to negative and positive resonance frequency $\pm\Omega_0$, respectively. Right panel: same as left panel but for including additional modulations of the hopping $\delta t=-0.24t$.[]{data-label="fig3"}](Fig3.eps){width=".9\columnwidth"}

 Z impurity
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The LDOS pattern predicted for an unitary scatterer in a $d$-wave superconductor with noninteracting quasiparticles is well-known to have crucial qualitative differences with the measured STS conductance maps $G(eV,\rr)$, at least if these are interpreted in the usual way as being directly proportional to the LDOS. The primary discrepancy is the existence of an observed intensity maximum on the central site of the impurity pattern in the case of Zn, which is impossible in the naïve theory for potentials $V_{3}$ larger than the bandwidth, since 