  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.
