  Various unconventional isotope effects have been reported in high-temperature
superconducting copper oxides which are beyond the scheme of BCS theory. Their
origin is investigated within polaron theory which leads to a renormalization
of the single particle energies and introduces a level shift here. It is found
that the exponential squeezing of the second nearest neighbour hopping integral
carries the correct isotope effect on the superconducting transition
temperature Tc, as well as the one on the penetration depth. The average
superconducting gap is predicted to have an isotope effect comparable to the
one on the penetration depth. The results imply that the coupling of the
electronic degrees of freedom to the Jahn-Teller Q2-type mode is the origin of
these isotope effects.
