Rescaled perturbation theory

A nonperturbative method that can go beyond the weak coupling perturbation theory is introduced. The essential idea is to formulate a set of exact differential equations as a function of coupling strength g. Unlike other resummation methods in which information on higher-order terms is necessary, we only need a leading-order perturbative formula in every step to reach a large value of g. The method is applied to the quantum anharmonic oscillator and quantum double-well potential in one dimension. Both are known to have divergent series in the weak coupling perturbation and the latter is not Borel-summable. Our method is shown to work well from the weak coupling to the strong coupling for the energy eigenvalues and the wave functions. The method is also applied successfully to a system with a time-dependent external field.