Nonperturbative Quantum Physics from Low-Order Perturbation Theory

The Stark effect in hydrogen and the cubic anharmonic oscillator furnish examples of quantum systems where the perturbation results in a certain ionization probability by tunneling processes. Accordingly, the perturbed ground-state energy is shifted and broadened, thus acquiring an imaginary part which is considered to be a paradigm of nonperturbative behavior. Here we demonstrate how the low order coefficients of a divergent perturbation series can be used to obtain excellent approximations to both real and imaginary parts of the perturbed ground state eigenenergy. The key is to use analytic continuation functions with a built-in singularity structure within the complex plane of the coupling constant, which is tailored by means of Bender-Wu dispersion relations. In the examples discussed the analytic continuation functions are Gauss hypergeometric functions, which take as input fourth order perturbation theory and return excellent approximations to the complex perturbed eigenvalue. These functions are Borel consistent and dramatically outperform widely used Padé and Borel-Padé approaches, even for rather large values of the coupling constant.

Figure 1

Real $\mathrm{\Delta }(F)$ and imaginary $\mathrm{\Gamma }(F)$ part of the perturbed ground state energy of: (a) the Stark Hamiltonian as a function of the electric field strength $F$; (b) the anharmonic oscillator with real perturbation $F{x}^3$; and (c) the anharmonic oscillator with imaginary perturbation $iF{x}^3$. We compare numerically exact values (dots) [30, 31, 34], with the fourth-order hypergeometric approximant ${}_2{F}_1$ (solid line) and Padé approximants (dashed line). In all three cases the ${}_2{F}_1$ approximant improves over Padé approximants [of the same-order in panels (a) and (b); and of much higher order in panel (c)] for the calculation of both $\mathrm{\Delta }(F)$ and $\mathrm{\Gamma }(F)$.

Figure 2

Panels (a)–(c) are as in Fig. 1, but calculated using the fourth-order hypergeometric approximant ${}_2{F}_1$ (solid line), Borel-hypergeometric method (dashed line) in Eq. (5), and the numerically exact values taken from the literature (filled dots) [30, 31, 34]. In panel (a) we also show the Padé-Borel results (empty dots) from Ref. [17]. The Borel-hypergeometric and hypergeometric methods are in excellent agreement, both yielding excellent approximations to both $\mathrm{\Delta }(F)$ and $\mathrm{\Gamma }(F)$ in all three cases.

Figure 3

Imaginary part of ${}_2{F}_1(h_1,h_2,h_3,h_{4}f)$ calculated for the Stark Hamiltonian in the complex $F$ plane. The built-in branch cut extends from $(h_4{)}^{-1}$ to $\infty$, and is essential to obtain $\mathrm{\Gamma }(F)\ne 0$. The $h_i$ are determined from the first four coefficients of the perturbation expansion. These yield $h_4\approx 164961$.