We show that in anisotropic superconductors, falling at finite temperature into the three-dimensional $\mathrm{XY}$ (3D-$XY)$ and at zero temperature into the two-dimensional $\mathrm{XY}$ quantum superconductor to insulator transition (2D-$\mathrm{XY}$-QSI) universality class, the isotope effects on transition temperature, specific heat, and magnetic penetration depths are not independent, but related by universal relations. The corresponding experimental data for cuprate superconductors reveals suggestive consistency with these relations. They imply a dominant role of fluctuations so that pair formation and pair condensation do not occur simultaneously. For this reason and due to the occurrence of the 2D-$\mathrm{XY}$-QSI transition the isotope effects do not provide information on the underlying pairing mechanism but must be attributed to a shift of the phase diagram upon isotope substitution. The 2D-QSI transition is driven by the exchange of pairs, which favors superconductivity and the combined effects of disorder and Coulomb repulsion of the pairs, which favor localization. Since isotope substitution will hardly affect disorder and Coulomb repulsion, the shift of the phase diagram must be attributed to the electron-phonon interaction, which renormalizes the mass of the fermions and with that the mass and the exchange of the pairs.

• Received 31 October 2002

DOI:https://doi.org/10.1103/PhysRevB.67.134514