  We study Andreev states near atomic scale modulations in the pairing
potential in both $s$- and d-wave superconductors with short coherence lengths.
For a moderate reduction of the local gap, the states exist only close to the
gap edge. If one allows for local sign changes of the order parameter, however,
resonances can occur at energies close to the Fermi level. The local density of
states (LDOS) around such pairing potential defects strongly resembles the
patterns observed by tunneling measurements around Zn impurities in
Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ (BSCCO). We discuss how this phase impurity model
of the Zn LDOS pattern can be distinguished from other proposals
experimentally.
