High Orders of 1/D-Expansion for Three-Body Systems

Aleksey V. Sergeev

S.I.Vavilov State Optical Institute, Tuchkov per. 1, St.Petersburg, 199034 Russia

E-mail: SERGEEV@soi.spb.su

Within the large-dimensional framework, the quantum-mechanical problem reduces to a classical static problem and a subsequent vibrational analysis. For a variety of three-body systems, the expansion coefficients were calculated to high order in [1].

Here, the attention is paid to the behavior of high orders of the 1/D- expansion and to the pertinent summation procedures. Typically, the coefficients in the expansion grow as factorials, Ek~(c0ak+c0*a*k)kbk! with k**. Our earlier results [2] are extended to three-dimensional effective potentials for treating three-body systems. The parameter a is calculated as reciprocal of an action integral along the complex classical trajectory in an inverted potential. We hope to improve Borel summation method by taking into account square-root singularities of the Borel function, using Darboux- Borel approximants. As an example, two-electron atoms are studied in detail. We verify the previous estimates of singularities ds and dp, obtained by quadratic Padé analysis [3].

Further, we investigate the divergence of the 1/D-expansion for excited states, when near-degeneracy frequently occurs. The reason is the fact, that two or more different states (known as Fermi resonances) may have approximately the same energy V0+(n1+1/2)w1+(n2+1/2)w2+(n3+1/2)w3. Such states are highly sensible to the perturbation, originated from anharmonic terms, so the 1/D-expansion rapidly diverges. The example is 1s2s1S state of helium when the energy has a branching point at 1/D= -0.0114 [4]. Suitable modification of the summation procedure is proposed that avoids the troubles related to energy quasi-crossing.

[1] A.V.Sergeev, Yad. Fiz. 50 (1989) 945 (Sov. J. of Nucl. Phys.).

[2] V.S.Popov, A.V.Sergeev, Phys. Lett. A 193 (1994) 165.

[3] D.Z.Goodson, M.López-Cabrera et al, J. Chem. Phys. 97 (1992) 8481.

[4] D.Z.Goodson, D.K.Watson, Phys. Rev. A 48 (1993) 2668.

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Designed by A. Sergeev.