Padé summations for the real and imaginary parts of atomic stark eigenvalues

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Abstract
The problem of direct Padé summation of Rayleigh–Schrödinger perturbation coefficients for the H atom Stark effect is discussed in light of the location of the Padé representation of the expected Bender‐Wu branch cuts in the complex F (field) plane. The resulting conclusion that such direct summation of the divergent perturbation expansion [for the real part of E(F)] might give suggestive, but certainly not convergent, results is documented. The relationship between the Padé problem of representation of cuts and corresponding problems in L2‐discretized scattering theory is pointed out. An alternative Padé expansion with newly placed cuts is proposed which is expected to, and indeed does, have excellent convergence properties for real F. The real and imaginary parts of the Stark broadened and shifted hydrogenic ground state are thus obtained directly from the (real) Rayleigh–Schrödinger perturbation coefficients for a large range of fields indicating the power and simplicity of the method.