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The recursive Neville algorithm allows one to calculate interpolating functions recursively. Upon a judicious choice of the abscissas used for the interpolation (and extrapolation), this algorithm leads to a method for convergence acceleration. For example, one can use the Neville algorithm in order to successively eliminate inverse powers of the upper limit of the summation from the partial sums of a given, slowly convergent input series. Here, we show that, for a particular choice of the abscissas used for the extrapolation, one can replace the recursive Neville scheme by a simple one-step transformation, while also obtaining access to subleading terms for the transformed series after convergence acceleration. The matrix-based, unified formulas allow one to estimate the rate of convergence of the partial sums of the input series to their limit. In particular, Bethe logarithms for hydrogen are calculated to 100 decimal digits. Generalizations of the method to series whose remainder terms can be expanded in terms of inverse factorial series, or series with half-integer powers, are also discussed.
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Logarithmic terms in atom-surface potentials: Limited applicability of rational approximations for intermediate distance
It is usually assumed that interaction potentials, in general, and atom-surface potential, in particular, can be expressed in terms of an expansion involving integer powers of the distance between the two interacting objects. Here, we show that, in the short-range expansion of the interaction potential of a neutral atom and a dielectric surface, logarithms of the atom-wall distance appear. These logarithms are accompanied with logarithmic sums over virtual excitations of the atom interacting with the surface in analogy to Bethe logarithms in quantum electrodynamics. We verify the presence of the logarithmic terms in the short-range expansion using a model problem with realistic parameters. By contrast, in the long-range expansion of the atom-surface potential, no logarithmic terms appear, and the interaction potential can be described by an expansion in inverse integer powers of the atom-wall distance. Several subleading terms in the large-distance expansion are obtained as a byproduct of our investigations. Our findings explain why the use of simple interpolating rational functions for the description of the atom-wall interaction in the intermediate regions leads to significant deviations from exact formulas.
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- Award ID(s):
- 2110294
- PAR ID:
- 10530187
- Publisher / Repository:
- American Physical Society
- Date Published:
- Journal Name:
- Physical Review A
- Volume:
- 108
- Issue:
- 1
- ISSN:
- 2469-9926
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1103/PhysRevA.108.012815
