A tractable example of perturbation theory with a field cutoff: the anharmonic oscillator

For λΦ4 models, the introduction of a large field cutoff improves significantly the accuracy that can be reached with perturbative series but the calculation of the modified coefficients remains a challenging problem. We show that this problem can be solved numerically, and analytically in the limits of large and small field cutoffs, for the ground-state energy of the anharmonic oscillator. For the two lowest orders in λ, the approximate formulae obtained in the large field cutoff limit extend unexpectedly far in the low field cutoff region and there is a significant overlap with the small field cutoff approximation. For the higher orders, this is not the case; however the shape of the transition between the small field cutoff regime and the large field cutoff regime is approximately order independent