Convergent perturbative series via finite path integral limits: application to energy at strong coupling of the anharmonic oscillator

Solving quantum field theories at strong coupling remains a challenging task. The main issue is that the usual perturbative series are asymptotic series which can be useful at weak coupling but break down completely at strong coupling. However, we show that if the limits of integration in the path integral are finite, the perturbative series is actually an absolutely convergent series which works well at strong coupling. For now, we apply this perturbative approach to λ ϕ⁴ theory in 0 + 0 dimensions (a basic integral) and 0+1 dimensions (quantum anharmonic oscillator). We begin by showing that finite integral limits yield a convergent series in agreement with exact analytical results for a basic integral.  We then consider the energy of the anharmonic oscillator whose perturbative series expansion in powers of the coupling fails at strong coupling. In quantum mechanics, if one is interested in the energy, it is often easier to use Schrödinger’s equation to develop a perturbative series than path integrals. Finite path integral limits are then equivalent to placing infinite walls at positions −L and L in the potential where L is positive, finite and can be arbitrarily large. With walls, the series expansion for the energy is now convergent and approaches the energy of the anharmonic oscillator as the walls are moved further apart. We use the convergent series to calculate the ground state energy at strong coupling and the result agrees with the exact (correct) energy to within 0.1%, a remarkable result in light of the fact that at strong coupling the usual perturbative series diverges badly right from the start.