  There is growing evidence that the unconventional spatial inhomogeneities in
the doped high-Tc superconductors are accompanied by the pairing of electrons,
subsequent quantum phase transitions (QPTs), and condensation in coherent
states. We show that these superconducting states can be obtained from phase
separation instabilities near the quantum critical points. We examine electron
coherent and incoherent pairing instabilities using our results on exact
diagonalization in pyramidal and octahedron Hubbard-like clusters under
variation of chemical potential (or doping), interaction strength, temperature
and magnetic field. We also evaluate the behavior of the energy gap in the
vicinity of its sign change as a function of out-of-plane position of the
apical oxygen atom, due to vibration of apical atom and variation of inter-site
coupling. These results provide a simple microscopic explanation of
(correlation induced) supermodulation of the coherent pairing gap observed
recently in the scanning tunneling microscopy experiments at atomic scale in
$Bi_2Sr_2CaCu_2O_{8+\delta}$. The existence of possible modulation of local
charge density distribution in these materials is also discussed.
