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

Abstract

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 Hϭp 2 ϩx 2 ϩ␤x 2m with mу2. 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 ␤͓0,ϱ).

Renormalized Perturbation Theory for Quartic Anharmonic Oscillator

“…The results of [7,8] show that, in contrast to the expansions (3), (4), and (9), the expansion (10) converges for all } # (0, 1], i.e., for all ;>0. The transformation described by Eqs.…”

“…From this analytic structure, exact dispersion relations for the energy E R (}) and c n and 1 n coefficients are found. It is shown that the large-order formulas for the c n and 1 n coefficients found in previous papers [2,5,7,8,11] follow simply from these dispersion relations. The summation rules for the 1 n coefficients are also discussed.…”

“…Later, it was derived from the large-order formula for the c n coefficients (53) [8]. Our approach represents direct and more rigorous derivation without using the large-order formula for the c n coefficients.…”

“…analytic at least for } min |}| <} max , where } min and } max correspond to ; min via Eq. (8). The values of } min and } max can be determined as follows.…”

“…It was shown in [10] that, except for a numerical factor, the transformation (7) (8) is the only transformation between E(;) and E R (}) for which the convergence of the DE ODM can be proven. We note that instead of this rather mathematical approach other, physically motivated arguments were used in [4] to introduce the renormalization (7) (8).…”