7 As we have mentioned above, in the framework of ki- netic energy driven SC mechanism, the electron Cooper pairs originating from the d-wave charge carrier pairing state are due to the charge-spin recombination, therefore there is a coexistence of the electron Cooper pair and short-range AF correlation, and hence the short-range AF fluctuation can persist into the SC state $^{17}$. In par- ticular, this charge-spin recombination is characterized by a convolution of the spin Green’s function and charge carrier Green’s function $^{17}$. The main consequence of this convolution is that the hole-like charge carrier d-wave BCS formalism is transferred into the electron-like d- wave BCS formalism $^{25}$. In other words, main difference between the hole-like charge carrier Green’s function (6) and the electron-like Green’s function (7) is a shift of the momentum by the AF wave vector Q = [ π, π ] in the Brillouin zone $^{25}$. This leads to a fact that the quasi- particle states near the antinodal points are broadened. Moreover, these quasiparticle states around the antin- odal regime are further suppressed due to the presence of the impurity scattering potential (9). On the other hand, the kinetic energy driven d-wave SC state is con- trolled by both the SC gap function and the SC quasipar- ticle coherent weight, which indicates that only coherent Bogoliubov quasiparticles at the Fermi surface are avail- able for superconductivity since everything happens at the Fermi surface. In this case, although the Bogoliubov quasiparticles can be excited by either positive or nega- tive energies $^{35}$, the excitation spectrum is a particle-hole symmetric thin Dirac cone around the nodal regime, then QSI below the high energy and its extinction at the high energy occur in this case, where this high energy which sets the QSI extinction is determined not only by the d-wave SC gap function alone but also by the coherent Bogoliubov quasiparticles located on the thin Dirac cone. In summary, within the framework of the kinetic en- ergy driven SC mechanism, we have discussed the QSI phenomenon in cuprate superconductors in the SC state by considering the single point-like impurity scattering potential. This QSI is characterized by the peaks in LDOS or the ratio of LDOS at opposite energy. By calculation of the momentum and energy dependence of the Fourier transformed LDOS δρ ( q , ω ) and the Fourier transformed ratio of LDOS at opposite energy δZ ( q , ω ), we have shown that the remarkable QSI phenomenon ob- served from STM experiments on cuprate superconduc- tors can be described qualitatively by a single point-like impurity in the kinetic energy driven homogeneous d- wave SC state. The amplitude of the peak increases with increasing energy at the low energy, and reaches a maxi- mum at the intermediate energy, then diminishes to zero at the high energy. The theory also predicts that with increasing doping concentration, the position of the peak along the nodal direction moves towards to the center of the Brillouin zone, while the position of the peak along the antinodal direction is shifted to large momentum re- gion, which should be verified by further STM experi- ments. Within the framework of the kinetic energy driven SC mechanism $^{17}$, our present results of QSI below the high The authors would like to thank H. S. Zhao for the helpful discussions. This work was supported by the National Natural Science Foundation of China under Grant No. 10774015, and the funds from the Ministry of Science and Technology of China under Grant Nos. 2006CB601002 and 2006CB921300. energy and its extinction at the high energy due to the presence of the impurity scattering potential (9) are qual- itatively consistent with the recent STM experimental data $^{4,5,14}$. Establishing this agreement is important to confirming the nature of a single impurity in the kinetic energy driven homogeneous d-wave SC state in cuprate superconductors. Although a quantitative understanding of the QSI phenomenon is not straightforward to obtain, and depends rather sensitively on the nature of the scat- tering medium, in this paper we are primarily interested in exploring the general notion of the QSI phenomenon induced by a single impurity in the kinetic energy driven homogeneous d-wave SC state. The qualitative agree- ment between the present theoretical results and STM experimental data also show that the presence of im- purities plays a crucial role for the QSI phenomenon in cuprate superconductors in the SC state. Although this QSI extinction at the high energy 14 also can be fitted by using a phenomenological d-wave BCS formalism with the same imaginary part of the self-energy (19), how- ever, no explicit calculations of the doping dependence of | δZ ( q , ω ) | , as shown in Fig. 3, can be made within a phenomenological d-wave BCS formalism. Under the kinetic energy driven SC mechanism, the external magnetic field aligns the spins of the unpaired electrons, then the singlet charge carrier pairs can not take advantage of the lower energy offered by a spin- polarized state. In this case, the magnetic field depen- dence of the penetration depth and superfluid density have been studied recently $^{36}$, where the superfluid den- sity decreases with increasing magnetic field, in agree- ment with the experimental results $^{37}$. With this study$^{36}$, we therefore can predict that QSI should be magnetic field dependent, and then the energy value of the QSI extinction decreases with increasing magnetic field. One of the typical features of d-wave superconductivity in cuprate superconductors is the particle-hole symmetric octet of dispersive Bogoliubov quasiparticle interference modulations $^{3–5}$. However, the recent STM experiments38 have reported on this octet’s evolution from low temper- atures to well into the underdoped pseudogap regime, where no pronounced changes occur in the octet phe- nomenology at the superconductor’s critical temperature T$_{c}$ , and it survives up to at least temperature T ∼ 1 . 5 T$_{c}$ . Thus an important issue is how this octet’s evolution from the SC state into the pseudogap regime is fitted within the framework of the kinetic energy driven su- perconductivity. These and the related issues are under investigation now. Acknowledgments