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The potential problem of anharmonic lattice vibrations in high-temperature superconductors (HTS) using the most suitable Born–Mayer–Huggins (BMH) potential has been taken up to investigate the renormalized phonon density of states (RPDOS). In order to develop the suitable results, the many body theory using quantum dynamical approach of Green’s function (GF) via an almost complete Hamiltonian (without using BCS type Hamiltonian) has been incorporated which includes harmonic electron and phonon Hamiltonian, electron–phonon interaction Hamiltonian, anharmonic and defect Hamiltonian. The derivation of anharmonic phonon GF for modified form of BMH potential enables to evaluate the renormalized and perturbed mode phonon frequencies of solids. The expressions obtained for the RPDOS can be resolved into diagonal and nondiagonal parts of which the nondiagonal part chiefly depends on impurities and disappears in pure crystals. A large number of new features are investigated in the light of this formulation followed by the numerical analysis for the energy spectrum of representative cuprate HTS YBa2Cu3O[Formula: see text]. The inclusion of defects, anharmonicities and electron–phonon interactions in the cuprate superconductors substantially modifies the scenario of RPDOS and reveals a large number of unexplained peaks in the spectrum.

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Considering Born–Mayer–Huggins potential as a most suitable potential to study the dynamical properties of high-temperature superconductors (HTS), the many-body quantum dynamics to obtain phonon Green’s functions has been developed via a Hamiltonian that incorporates the contributions of harmonic electron and phonon fields, phonon field anharmonicities, defects and electron–phonon interactions without considering BCS structure. This enables one to develop the quasiparticle renormalized frequency dispersion in the representative high-temperature cuprate superconductor YBa2Cu3O[Formula: see text]. The superconducting gap shows substantial changes with increased doping. The in-plane gap study revealed a [Formula: see text]-shape gap with a nodal point along [Formula: see text] direction for optimum doping ([Formula: see text] = 0.16) and the nodal point vanished in underdoped and overdoped regimes. The d[Formula: see text] pairing symmetry is observed at optimum doping with the presence of s or d[Formula: see text] components ([Formula: see text] 3%) in underdoped and overdoped regimes.