The Inward Dispersion of the Neutron Scattering Experiments in HTSC   Cuprates
  The theory of the high temperature superconducting cuprates, which is based on the condensation of holes into strings in checker-board geometry, was successful to explain the elastically scattered Neutrons by spin waves. Here it is extended to analyze the inward dispersion curve of its inelastic counterpart, up to the resonance energy- . This extension is done by applying the perturbation theory of the linear response to the condensed strings. The approximated susceptibility is derived by means of the ring diagram. The dispersion relation is obtained from the dispersion of the poles of the susceptibility integral. It is found that the particle anti-particle pair that yields the susceptibility is the time reversal pair where the particle momentum is in phase A, and the anti-particle momentum is in phase B. The dispersion is found to be in agreement with experiment, subject to some suggested corrections. The weak intensity by the resonance energy, as well as the dispersion, is speculated to be modified due to interference with spin waves that are caused by direct spin flip, as in the mother undoped materials. 
Moshe Dayan
cond-mat.supr-con