  The issue of probing the pseudogap regime of the cuprate superconductors,
specifically with regard to the existence and nature of superconducting pairing
correlations of d-wave symmetry, is explored theoretically. It is shown that if
the d-wave correlations believed to describe the superconducting state persist
into the pseudogap regime, but with pair-potential phase-fluctuations that
destroy their long-range nature, then the low-energy quasiparticle states
observed near extended impurities in the truly superconducting state should
also persist as resonances in the pseudogap regime. The scattering of
quasiparticles by these phase-fluctuations broadens what was (in the
superconducting state) a sharp peak in the single-particle spectral function at
low energy, as we demonstrate within the context of a simple model. This peak
and its broadening is, in principle, accessible via scanning tunneling
spectroscopy near extended impurities in the pseudogap regime. If so, such
experiments would provide a probe of the extent to which d-wave superconducting
correlations persist upon entering the pseudogap regime, thus providing a
stringent diagnostic of the phase-fluctuation scenario.
