**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*~(*c*0*ak*+*c*0**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 d*s*
and d*p*, 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 *V*0+(*n*1+1/2)w1+(*n*2+1/2)w2+(*n*3+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.

Back to Presentations on conferences.

Designed by A. Sergeev.