An Alternative Perturbative Expansion in Quantum Mechanics. Scaling and cut-off Resummation

An alternative perturbative expansion in quantum mechanics which allows a full expression of the scaling arbitrariness is introduced. This expansion is examined in the case of the anharmonic oscillator and is conveniently resummed using a method which consists in introducing an energy cut-off that is carefully removed as the order of the expansion is increased. We illustrate this technique numerically by computing the asymptotic behavior of the ground state energy of the anharmonic oscillator for large couplings, and show how the exploitation of the scaling arbitrariness substantially improves the convergence of this perturbative expansion.