Superconducting phase coherence in the presence of a pseudogap: Relation   to specific heat, tunneling and vortex core spectroscopies
  In this paper we demonstrate how, using a natural generalization of BCS theory, superconducting phase coherence manifests itself in phase insensitive measurements, when there is a smooth evolution of the excitation gap \Delta from above to below Tc. In this context, we address the underdoped cuprates. Our premise is that just as Fermi liquid theory is failing above Tc, BCS theory is failing below. The order parameter \Delta_{sc} is different from the excitation gap \Delta. Equivalently there is a (pseudo)gap in the excitation spectrum above Tc which is also present in the underlying normal state of the superconducting phase, and can be directly inferred from specific heat and vortex core experiments. At the same time many features of BCS theory, e.g., fermionic quasiparticles below Tc, are clearly present. These observations can be reconciled by a natural extension of BCS theory, which includes finite center-of-mass momentum pair excitations, in addition to the usual fermionic quasiparticles. Applying this theory we find that the Bose condensation of Cooper pairs, which is reflected in \Delta_{sc}, leads to sharp peaks in the spectral function once $T \le T_c$. These are manifested in ARPES spectra as well as in specific heat jumps, which become more like the behavior in a \lambda transition as the pseudogap develops. We end with a discussion of tunneling experiments and condensation energy issues. Comparison between theoretical and experimental plots of C_v, and of tunneling and vortex core spectroscopy measurements is good. 
Qijin Chen, K. Levin, and Ioan Kosztin