Doping dependence of density response and bond-stretching phonons in cuprates

We explain the anomalous doping dependence of zone boundary $(\pi,0)$ and $(\pi,\pi)$ bond-stretching phonons in the high-temperature superconductor La$_{2-\delta}$Sr$_{\delta}$CuO$_4$ in the doping range $0<\delta<0.35$. Our calculations are based on a theory for the density response of doped Mott-Hubbard insulators.