| In the theoretical analyses of impurity effects in superconductors the assumption is usually made that all quantities, except for the Green functions, are slowly varying functions of energy. When this so-called Fermi Surface Restricted Approximation is combined with the assumption that impurities can be represented by $\delta$-function potentials of arbitrary strength, many reasonable looking results can be obtained, even though elementary examples show that $\delta$-function potentials do not lead to scattering at all in systems of two and more dimensions. The generalization to finite range potentials appears to be straightforward. However, the selfenergy resulting from scattering off finite range impurities of infinite strength, such as hard spheres, diverges in this approximation at frequencies much larger than the gap amplitude! To avoid this unacceptable result, the energy dependencies of all quantities involved has to be properly taken into account. This requires the selfconsistent solution of coupled twodimensional integral equations. We present results for a single impurity $t$-matrix and ensemble averaged selfenergy corrections for the normal state and a $d$-wave superconductor. The changes in the density of mid-gap states as compared to $\delta$-function scatterers are presented. We shall focus, however, on the spectral function, as measured in ARPES, which reflects the range and the strength of the impurity potential most sensitively and which no longer shows particle-hole symmetry. |
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