Borel summability: Application to the anharmonic oscillator

Numerical comparisons of nonlinear convergence accelerators

“…Different methods of evaluation of the same quantities can produce noticeably different numerical results, especially when transforming rapidly divergent series. The 6 and u results are better than those from e (or Padé) in every case, and they are nearly as good as the Borel-Padé results in [9]. Graffi, et al, give rate of convergence data only for ß -OA and 1.0 (their Table 1).…”

“…Bender and Wu [3] have computed the first 75 terms of the Rayleigh-Schrödinger series for the ground state energy £(1, ß) of the anharmonic oscillator. Simon [18] has reported extensive computations with this series using the diagonal of the Padé table (which is theoretically equivalent to using the e algorithm [17], [23]), and Graffi, et al [9] have obtained more accurate results by a Borel-Padé summation technique. In particular, Tables 2 and 3 in [9] give the most accurate values available for E(\, ß), ß = 0.1(0.1)1.0 and ß = 1(1)14.…”

“…Simon [18] has reported extensive computations with this series using the diagonal of the Padé table (which is theoretically equivalent to using the e algorithm [17], [23]), and Graffi, et al [9] have obtained more accurate results by a Borel-Padé summation technique. In particular, Tables 2 and 3 in [9] give the most accurate values available for E(\, ß), ß = 0.1(0.1)1.0 and ß = 1(1)14. We have computed the effect of accelerating the series by e, 6, and u for ß = 0.1, 0.5, 1, 5, and 10, to compare the use of these simpler accelerators to the Borel-Padé summations.…”

On functions uniquely determined by their asymptotic expansion

“…However, one of the first reappearances of Watson's theorems occurred ten years earlier in connection with the theory of anharmonic oscillators, see [7].…”

Schrödinger operators in the twentieth century

“…See Simon 252 for a study of multiwell problems. In some cases, including ͑XII.2͒, it is known that the divergent perturbation series can be made to give the right eigenvalue with a summability method, either Padé approximation ͑Loeffel et al 187 ͒ or Borel summation ͑Graffi et al 102 ͒. Borel summability is also known to work for the Zeeman series for hydrogen-hydrogen perturbed by turning on a constant magnetic field; see Avron et al 21 and Avron et al 17 In certain cases, eigenvalues are perturbed into resonances, the subject of Sec.…”