Renormalized perturbation theory for quartic anharmonic oscillator

The analytic structure of the renormalized energy of the quartic anharmonic oscillator described by the Hamiltonian H = p(2) + x(2) + beta x(4) is dis cussed 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 ail positive values of the coupling consta nt beta but also for some double-well problems. Further? exact dispersion r elations for the weak and strong coupling expansion coefficients of the ren ormalized energy are derived. The large-order formulas for these coefficien ts found in previous papers Follow simply from the dispersion relations. Th e renormalized weak coupling expansion is separated into the Stieltjes and non-Stieltjes parts. Numerical tests performed for the ground and first exc ited states confirm correctness of our conclusions. Finally, properties of different perturbative approaches to the anharmonic oscillator are compared . (C) 1999 Academic Press.