The analytic structure of the renormalized energy of the quartic anharmonic oscillator described by the Hamiltonian Hamiltonian H=p~2 +x~2 +#beta#x~4 is discussed and the dispersion relation for the renormalized energy is found. It follows from the analytic structure that the renormalized strong coupling expansion converges not only for all positive values of the coupling constant #beta# but also for some double-well problems. Further, exact dispersion relations for the weak and strong coupling expansion coefficients of the renormalized energy are derived. The large-order formulas for these coefficients found in previous papers follow simply from the dispersion relations. The renormalized weak coupling expansion is separated into the Stieltjes and non-Stieltjes parts. Numerical tests performed for the ground and first excited states confirm correctness of our conclusions. Finally, properties of different perturbative approaches to the anharmonic oscillator are compared.
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