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In order to clarify that the Cooper pair in $\beta$-pyrochlore oxides ismediated by anharmonic oscillation of guest atom, i.e., rattling, we propose anexperiment to detect anomalous isotope effect. In the formula of $T_{\rm c}\propto M^{-\eta}$, where $T_{\rm c}$ is superconducting transition temperatureand $M$ denotes mass of the oscillator, it is found that the exponent $\eta$ isincreased with the increase of anharmonicity of a potential for the guest atom.We predict that $\eta$ becomes larger than 1/2 in rattling-inducedsuperconductor, in sharp contrast to $\eta=1/2$ for weak-couplingsuperconductivity due to harmonic phonons and $\eta<1/2$ for strong-couplingsuperconductivity with the inclusion of the effect of Coulomb interaction.

Anomalous Isotope Effect in Rattling-Induced Superconductor