Large-order behavior of the convergent perturbation theory for anharmonic oscillators

Using the large-order formula for the coefficients of the divergent weak-coupling series for the energy of the anharmonic oscillators, we derive a simple analytic large-order formula for the coefficients of the convergent renormalized strong-coupling series. This formula is valid for all the states of the anharmonic oscillators defined by the Hamiltonians H=p(2)+x(2)+beta x(2m) with m greater than or equal to 2. A further generalization of this formula is also proposed. Numerical tests of the formula are performed for the quartic, sextic, octic, and decadic oscillator with the help of asymptotic analysis. Further it is shown that the renormalized strong-coupling perturbation expansion converges for all the states of these oscillators and for all physically relevant beta epsilon [0,infinity). [S1050-2947(99)04901-X].