Softening of Cu-O bond stretching phonon in tetragonal   HgBa$_2$CuO$_{4+\delta}$
  Phonons in nearly optimally doped HgBa$_2$CuO$_{4+\delta}$ were studied by inelastic X-ray scattering. The dispersion of the low energy modes is well described by a shell model, while the Cu-O bond stretching mode at high energy shows strong softening towards the zone boundary, which deviates strongly from the model. This seems to be common in the hole-doped high-$T_\mathrm{c}$ superconducting cuprates, and, based on this work, not related to a lattice distortion specific to each material. 
H. Uchiyama, A.Q.R. Baron, S. Tsutsui, Y. Tanaka, W.-Z. Hu,   A.Yamamoto, S. Tajima, and Y. Endoh