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

Abstract

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. In this work, we show that if the limits of integration in the path integral are finite, the perturbative series is remarkably an absolutely convergent series which works well at strong coupling. For now, we apply this perturbative approach to $λφ^4$ theory in 0+0 dimensions (a basic integral) and 0+1 dimensions (quartic anharmonic oscillator). As a further application, we also consider the sextic anharmonic oscillator. For the basic integral, we show that finite integral limits yields a convergent series whose values are in agreement with exact analytical results at any coupling. This worked even when the asymptotic series was not Borel summable. It is well known that the perturbative series expansion in powers of the coupling for the energy of the anharmonic oscillator yields an asymptotic series and hence 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 weak, intermediate and strong coupling. At strong coupling, the result from the series agrees with the exact energy to within $0.1\%$, a remarkable result in light of the fact that at strong coupling the usual perturbative series diverges badly immediately.

Figures (22)

• Figure 1: Results at small (weak) coupling $\lambda=0.02$. The ground state energy $E_n$ at order $n$ is quoted to eight decimal places. The table contains the $\%$ error between the energy $E_n$ and the exact energy $E=0.51408643$ obtained numerically and quoted also to eight decimal places at the top of the table. The $\%$ error is completely negligible over a long range of orders (from roughly $n=5$ to $n=35$). This is the long plateau region on the plot and implies that the series can be used to make reliable predictions. After the plateau region, the series begins to diverge and reaches a very large $\%$ error of order $10^4$ at $n=50$. We therefore have an asymptotic series but because the coupling is small, this becomes only apparent starting at very large orders (roughly starting at order $n=40$).
• Figure 2: Results at intermediate coupling $\lambda=0.1$. The $\%$ error between the series $E_n$ and the exact value $E=0.55914633$ starts at a reasonably low value of $2.84\%$, dips to a minimum of $1.86 \%$ over the next two orders and quickly diverges afterwards (asymptotic series). So the series gets reasonably close to the exact value early on (at low orders) but in contrast to the weak coupling case, there is no plateau region where it settles close to the exact value over a long range of orders. This is why we refer to it as an "intermediate" coupling. In the next section, we obtain an absolutely convergent series that settles/converges to the correct value with negligible $\%$ error all the way to large order $n$.
• Figure 3: Results at large (strong) coupling $\lambda=0.2$. The $\%$ error starts at $8\%$ but only increases afterwards. The series never gets close to the exact value and then diverges from it very rapidly. We see that at strong coupling, the perturbative series breaks down completely and a new approach is required. It the next section we develop an absolutely convergent perturbative series which yields excellent results at strong coupling.
• Figure 4: Plot of the Kummer confluent hypergeometric function ${}_1F_1(-\frac{h}{4},\frac{1}{2}, x^2\,)$ as a function of $x$ for $h=0.001$. The function is basically unity until it drops off on both sides to zero at $x\approx \pm 3$.
• Figure 5: Results at weak coupling $\lambda=0.02$ with $h=0.005$. The series converges to the exact value of the energy to within less than $0.5\%$ up to arbitrary large orders (we show it here up to $n=50$). It is an absolutely convergent series. In contrast, the usual perturbative series at weak coupling diverges at large order as in the plot of Fig. \ref{['Lambda002']}.