A Tractable Example of Perturbation Theory with a Field Cutoff: the Anharmonic Oscillator

TL;DR

This paper explores a numerical and analytical approach to perturbation theory with a field cutoff in the anharmonic oscillator, revealing how cutoff size affects the accuracy of ground state energy calculations.

Contribution

It provides a tractable method to compute modified perturbative coefficients for the anharmonic oscillator with a field cutoff, including analytical limits and transition behavior.

Findings

Large cutoff formulas extend well into low cutoff regions

Significant overlap between large and small cutoff approximations at low orders

Transition shape between regimes is approximately order independent

Abstract

For lambda phi^4 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 lambda, the approximate formulas 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.