  One of the most prominent problems in high temperature superconductivity is
the nature of the pseudogap phase in underdoped regimes; particularly important
is the role of phase fluctuations. The Gutzwiller-projected BCS Hamiltonian is
a useful model for high temperature superconductivity due to an exact mapping
to the Heisenberg model at half filling and generally a very close connection
to the t-J model at moderate doping. We develop the dynamical mean field theory
for the d-wave BCS Hamiltonian with on-site repulsive interaction, $U$,
physically imposing the partial Gutzwiller projection. For results, two
pseudogap energy scales are identified: one associated with the bare pairing
gap for the singlet formation and the other with the local phase coherence. The
real superconducting gap determined from sharp coherence peaks in the density
of states shows strong renormalization from the bare value due to $U$.
