The convergence of large-order expansions in δ= 1/D, where D is the dimensionality of coordinate space, for energies E(δ) of Coulomb systems is strongly affected by singularities at δ= 1 and δ= 0. Padé-Borel approximants with modifications that completely remove the singularities at δ= 1 and remove the dominant singularity at δ= 0 are demonstrated. A renormalization of the interelectron repulsion is found to move the dominant singularity of the Borel function F(δ) = ∑jE′j/j!, where E′j are the the expansion coefficients of the energy with singularity structure removed at δ= 1, farther from the origin and thereby accelerate summation convergence. The ground-state energies of He and H+ 2 are used as test cases. The new methods give significant improvement over previous summation methods. Shifted Borel summation using Fm(δ) = ∑jE′j/Γ(j + 1 - m) is considered. The standard deviation of results calculated with different values of the shift parameter m is proposed as a measure of summation accuracy.
|Number of pages||11|
|Journal||Journal of Mathematical Physics|
|Publication status||Published - Oct 1998|
ASJC Scopus subject areas
- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics