  The isotope effect in the superconducting transition temperature is anomalous
if the isotope coefficient $\alpha<0$ or $\alpha>1/2$. In this work, we show
that such anomalous behaviors can naturally arise within the
Bardeen-Cooper-Schrieffer framework if both phonon and non-phonon modes
coexist. Different from the case of the standard Eliashberg theory (with only
phonon) in which $\alpha\le1/2$, the isotope coefficient can now take arbitrary
values in the simultaneous presence of phonon and the other non-phonon mode. In
particular, most strikingly, a pair-breaking phonon can give rise to large
isotope coefficient $\alpha>1/2$ if the unconventional superconductivity is
mediated by the lower frequency non-phonon boson mode. Based on our studies,
implications on several families of superconductors are discussed.
