On the Relationship Between the Pseudo- and Superconducting Gaps: Effects of Residual Pairing Correlations Below Tc

The existence of a normal state spectral gap in underdoped cuprates raises important questions about the associated superconducting phase. For example, how does this pseudogap evolve into its below Tc counterpart? In this paper we characterize this unusual superconductor by investigating the nature of the ``residual'' pseudogap below Tc and, find that it leads to an important distinction between the superconducting excitation gap and order parameter. Our approach is based on a conserving diagrammatic BCS Bose-Einstein crossover theory which yields the precise BCS result in weak coupling at any T<Tc and reproduces Leggett's results in the T=0 limit. We explore the resulting experimental implications.