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Rescaled Perturbation Theory

A non-perturbative method which can go beyond the weak coupling perturbation theory is introduced. Essential idea is to formulate a set of exact differential equations as a function of the coupling strength $g$. Unlike other resummation in which information of the higher order terms is necessary, we only need a leading order perturbative formula in every step to reach the 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 the system with time-dependent external field.