Optimized Rayleigh-Schrödinger Expansion of the Effective Potential

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An optimized Rayleigh-Schrödinger expansion scheme of solving the functional Schrödinger equation with an external source is proposed to calculate the effective potential beyond the Gaussian approximation. For a scalar field theory whose potential function has a Fourier representation in a sense of tempered distributions, we obtain the effective potential up to the second order, and show that the first-order result is just the Gaussian effective potential. Its application to the λϕ⁴ field theory yields the same post-Gaussian effective potential as obtained in the functional integral formalism.