We consider the resummation of the perturbation series describing the energy displacement of a hydrogenic bound state in an electric field (known as the Stark effect or the LoSurdo-Stark effect), which constitutes a divergent formal power series in the electric-field strength. The perturbation series exhibits a rich singularity structure in the Borel plane. Resummation methods are presented that appear to lead to consistent results even in problematic cases where isolated singularities or branch cuts are present on the positive and negative real axis in the Borel plane. Two resummation prescriptions are compared: (i) a variant of the Borel-Pad'e resummation method, with an additional improvement due to utilization of the leading renormalon poles, and (ii) a contour-improved combination of the Borel method with an analytic continuation by conformal mapping, and Pad'e approximations in the conformal variable. The singularity structure in the case of the LoSurdo-Stark effect in the complex Borel plane is shown to be similar to (divergent) perturbative expansions in quantum chromodynamics.
U. D. Jentschura, "Resummation of the Divergent Perturbation Series for a Hydrogen Atom in an Electric Field," Physical Review A. Atomic, Molecular, and Optical Physics, vol. 64, no. 1, pp. 134031-1340313, American Institute of Physics (AIP), Jul 2001.
The definitive version is available at https://doi.org/10.1103/PhysRevA.64.013403
Keywords and Phrases
Approximation Theory; Conformal Mapping; Electric Field Effects; Electron Energy Levels; Hydrogen; Integration; Mathematical Transformations; Numerical Analysis; Perturbation Techniques; Borel-Pade Resummation Method; Quantum Chromodynamics; Stark Effect; Quantum Theory
International Standard Serial Number (ISSN)
Article - Journal
© 2001 American Institute of Physics (AIP), All rights reserved.
01 Jul 2001