The influence of the critical fluctuations under the structural phase transition on the superconducting pairing

The influence of the critical fluctuations in the vicinity of the structural phase transition (SPT) on the isotope effect (IE) is studied within the anharmonic model of the high-temperature superconductivity. It is shown that if the superconducting transition temperature $T_c$ is equal to the temperature of SPT, then the IE is determined by the critical exponent $\gamma$. In the case $\gamma<1$ the isotope factor for $T_c$ $(\alpha=-\frac{\partial \ln T_c}{\partial \ln M}$) is equal to one for the structural transition temperature $T_0$ (i.e. $\alpha=\varepsilon$, where $\varepsilon=-\frac{\partial \ln T_0}{\partial \ln M}$), but in the case $\gamma>1$ $\alpha=\frac{1}{2}$ (if one neglects Coulomb interaction). This result does not depend on the model for the SPT. The result is demonstrated by considering the exactly solvable model of the SPT with $\gamma =2$.