4 This LDOS is closely related to the differential conduc- tance g ( r , ω ) since the result of the differential conduc- tance g ( r , ω ) is proportional to ρ ( r , ω ) at location r and energy ω =eV. However, the intense atomic-scale spatial fluctuations in electronic structure cause systematic er- rors in setting the STM tip elevation for the experimental measurement of the differential conductance g ( r , ω ) $^{12–14}$, therefore the ratio of differential conductances at oppo- site bias in Eq. (1) or its equivalent, the ratio of LDOS at opposite energy, has been measured experimentally for an enhancement of the QSI signatures $^{12–14}$. Since the condition ρ$_{0}$ ( ω ) ≫ δρ ( r , ω ) is well satisfied for cuprate superconductors, then the ratio of LDOS at opposite energy (15) can be obtained approximately as $^{33}$, where Z$_{0}$ ( ω ) = ρ$_{0}$ ( ω ) /ρ$_{0}$ ( − ω ), and only the first-order modulation δρ ( r , ± ω ) is kept, then the modulation of the ratio of LDOS at opposite energy can be expressed as, FIG. 1: (Color online) The Fourier transformed LDOS as a function of momentum in the full Brillouin zone with energy (a) ω = − 0 . 01 J , (b) ω = − 0 . 03 J , and (c) ω = − 0 . 05 J at tem- perature T = 0 . 002 J for the doping concentration δ = 0 . 15 in the presence of single point-like potential scatterer of strength V = 0 . 1 J (left panel). (d) to (f) are the corresponding results of the Fourier transformed ratio of LDOS at opposite energy (right panel). and δρ ( q , ω ) are clearly fourfold symmetric and display numerous local maxima (bright regions) at different q for different energies, where all of these q vectors are consis- tent with the prediction from the octet model $^{4,5}$. These bright regions in momentum-space display the intensity of the modulation for LDOS (Fig. 1a-c) or the intensity of the modulation for the ratio of LDOS at opposite en- ergy (Fig. 1d-f). In particular, the bright regions near the center of each figure in Fig. 1 a-c or Fig. 1d-f reflect long-wavelength inhomogeneity in δρ ( q , ω ) or δZ ( q , ω ), and are induced by the weak impurity scattering poten- tial (9). All these results are in qualitative agreement with the STM experimental results $^{4,5}$. and its Fourier transformation is evaluated explicitly as, It has been argued 12–14 that the observed QSI from δZ ( r , ω ) or δZ ( q , ω ) is an intrinsic phenomenon free from any set-point-related issues, which inevitably contami- nate g ( r , ω ) and g ( q , ω ). We are now ready to discuss the QSI phenomenon in cuprate superconductors in the SC state and its extinc- tion at the high energy. In cuprate superconductors, al- though the values of J and t is believed to vary somewhat from compound to compound, however, as a qualitative discussion, the commonly used parameters in this paper are chosen as t/J = 2 . 5 and t $^{′}$/t = 0 . 3. In this case, we have performed a calculation for the Fourier transformed LDOS δρ ( q , ω ) in Eq. (14) and the Fourier transformed ratio of LDOS at opposite energy δZ ( q , ω ) in Eq. (18), and the results of | δρ ( q , ω ) | as a function of momentum in the full Brillouin zone with energy (a) ω = − 0 . 01 J , (b) ω = − 0 . 03 J , and (c) ω = − 0 . 05 J at temperature T = 0 . 002 J for the doping concentration δ = 0 . 15 in the presence of single point-like potential scatterer of strength V = 0 . 1 J are plotted in Fig. 1 (left panel). For comparison, the corresponding results of | δZ ( q , ω ) | are also plotted in Fig. 1d-f (right panel). It is shown that the results obtained from δZ ( q , ω ) are almost the same as those of δρ ( q , ω ), reflecting an experimental fact that the measurement data from the ratio of LDOS at op- posite energy retain all the main information of QSI ob- served from LDOS. Moreover, both results from δZ ( q , ω ) To analyze this peak feature in Fig. 1 more clearly, we have made a series of calculations for the momen- tum dependence of δρ ( q , ω ) and δZ ( q , ω ) with different energies, and the results of | δρ ( q , ω ) | as a function of momentum along the (a) [0 , 0] → [ π, π ] (nodal) direction and (b) [0 , 0] → [ π, 0] (antinodal) direction with energy ω = − 0 . 01 J , ω = − 0 . 02 J , ω = − 0 . 03 J , ω = − 0 . 04 J , and ω = − 0 . 05 J (from bottom to top) at temperature T = 0 . 002 J for the doping concentration δ = 0 . 15 in the presence of single point-like potential scatterer of strength V = 0 . 1 J are plotted in Fig. 2 (left panel). For comparison, the corresponding results of | δZ ( q , ω ) | Z ( r , ω ) ≈ Z$_{0}$ ( ω ) [ 1 + δρ ( r , ω ) ρ$_{0}$ ( ω ) − δρ ( r , − ω ) ρ$_{0}$ ( − ω ) ] , (16) δZ ( r , ω ) = Z ( r , ω ) − Z$_{0}$ ( ω ) ≈ Z$_{0}$ ( ω ) [ δρ ( r , ω ) ρ$_{0}$ ( ω ) − δρ ( r , − ω ) ρ$_{0}$ ( − ω ) ] , (17) δZ ( q , ω ) ≈ Z$_{0}$ ( ω ) [ δρ ( q , ω ) ρ$_{0}$ ( ω ) − δρ ( q , − ω ) ρ$_{0}$ ( − ω ) ] . (18) Z ( r , ω ) = ρ ( r , ω ) ρ ( r , − ω ) , (15)