Borel sum

This singularity structure from the Borel sum differs somewhat from the singularity structure predicted from the coulombic poles of... 

Let us note that the series representing j(z) may not converge even for small z. Actually, there are reasons to believe that it diverges for all z. This can be understood by observing that a continuous theory with a purely attractive interaction (z < 0) does not exist 1 (see Chapter 15). However, it seems likely that the Borel transform of the series exists for small z and that j(z) can be defined for all z by analytic continuation of the sum of the transformed series (see Chapter 12). 

Boral sum An integral that is defined so as to represent the sum of a divergent series. The concept enables many of the divergent series that occur in physics, such as perturbation series in quantum field theory, to be mathematically well-defined. It is named after the French mathematician fimile Borel (1871-1956). 

It is obvious that an initial sum can be resummed in different ways. Apart from the Leroy fit parameter p mentioned above some arbitrariness arises from the different types of rational approximants one may construct. For instance, within the two-loop approximation the method of Pade-Borel resummation of a resolvent series can be done using either the [0/2] or the [1/1] approximants. The Chisholm-Borel approximation implies even more arbitrariness and demands a careful analysis of the approximants to be chosen. 

This assumption implies that the sum in this form converges to the trae (but unknown) value of the groimd-state eneigy. In most cases, this assumption is actually not satisfied. Starting at a certain order the correction can become laiger than the first-order contribution - the series diverges. Tmncating such a series after it has swapped phases can only yield reasonable results, if resummation methods (Fade, Borel, variational approaches, etc.) are applied which are based on the fact that the trae result must be finite. 

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