  We report a model study of the phonon response on the interplay of the
superconducting (SC) gap and both static and dynamic Jahn-Teller distortion and
predict the appearance of the SC and Jahn-Teller (JT) distortion gap excitation
peaks in the Raman spectra of high-Tc superconductors. The model consists of
the Hamiltonian containing static JT interaction and s-wave type SC interaction
in the conduction band. Further the phonons are coupled to the density of the
conduction electron in the band as well as to the JT split conduction band
giving rise to dynamic Jahn-Teller (DJT) interaction. The phonons are
considered in a harmonic approximation. The phonon Green's function is
calculated by Zubarev's technique and the phonon self-energy arising due to
normal electron-phonon (EP) interaction and DJT interaction. The phonon
self-energy arising due to this is calculated from the electron response
density function in the coexistence phase of the two order parameters. The
phonon spectral density predicts two excitation peaks, one due to SC gap and
the other due to JT distortion. The DJT coupling suppresses the JT gap while it
enhances SC gap as well as SC transition temperature. The evolution of these
excitation peaks are investigated by varying different model parameters of the
system.
