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We investigate a system consisting of strongly correlated localized and itinerant electron states mixed by optical vibration. The linear vibronic interaction causes the softening of active optical mode and it may induce a structural instability. The modification of renormalized phonon frequency by the gap in the energy of itinerant electrons in the superconducting phase has been established. The disposition of the electron spectrum and chemical potential influences substantially the corresponding effect. It is demonstrated that the presence of van Hove singularity in the center of itinerant electron band introduces qualitative changes into the superconductivity-induced shifts of phonon frequency.

Superconductivity-induced shift of phonon frequency in the system with itinerant and strongly correlated electrons