›1mLarge-order dimensional expansion for three-body systems›m


›3mS.I.Vavilov State Optical Institute,

Tuchkov per. 1, Saint Petersburg, 199034 Russian Federation,

e-mail sergeev@soi.spb.su›m

The energy of bound and quasistationary states of

quantum-mechanical few-body systems is expanded as a power series

in 1/N, where N is the dimensionality of the space. A

multidimensional continuation of the problem is performed, and N

is treated as a continuous parameter with an initial problem

corresponding to the physical value N = 3. When N is inflated to

infinity, the dynamics becomes classical. The limit

corresponds to the classical motion around the common centre of

masses of the rigid configuration of bodies in a plane or in

four-dimensional space for two or three-body systems,

correspondingly [1]. In this simplifying limit, the problem

reduces to minimization of an effective potential containing an

additional centrifugal term, for two-body system or

its analogue for three-body system,

where , and are the altitudes in a configurational


At finite dimensions the bodies undergo small oscillations

about fixed positions that correspond to the minima of the

effective potential. The 1/N-expansion is similar to the methods

of molecular vibration analysis. It reduces to the Rayleigh -

Schrödinger perturbation theory for anharmonic oscillator. The

expansion coefficients can be calculated exactly and to high

order, using recursive relations [1, 2].

The difficulties arise because of the divergence of the

expansion, which renders convential summation methods ineffective

beyond the lowest orders. To use the special summation methods,

such as Padé - Borel approximants, one should know the divergent

large-order behaviour of the expansion responsible for the

singularities in the Borel function [2]. Typically, the

coefficients in the expansion grow as factorials,

or with . To find the

parameters , and , the dispersion relations were used

which connect with the integral from the imaginary part of

the energy. Particularly, coincides with the action integral

standing in the exponent in the quasiclassical formula for decay


where is the minimum of the effective potential, and

is a turning point, [3]. For bound states, there

is a pair of complex-conjugate turning points, so the large-order

asymptotics contains two terms.

Here the earlier results [3] are extended to multidimensional

effective potentials for treating three-body systems. In this

case, we deal with a three-dimensional quantum decay problem, the

number of variables being equal to the number of interparticle

distances. The central problem is the solution of the eikonal

equation and minimization of the classical action in order to

determine the parameter . Two different approaches are used.

The first one is based on the method of characteristics. The

classical trajectories in an inverted effective potential are

calculated, a trajectory is chosen which terminates at a

stopping point and which represents the most probable escape path

[4]. The parameter equals to the reciprocal of the

action along this trajectory. In the second approach, the action

is expanded as a perturbation series around the minimum of the

effective potential. As an illustration we investigate the

dependence of the parameter on the nuclear charge for the

members of helium isoelectronic sequence. The results will be

incorporated into summation schemes in order to yield highly

accurate energies.

1. A.V.Sergeev,Yadernaya Fizika ›1m50›m,945(1989)[Sov.J.of Nucl.Phys].

2. D.Z.Goodson, M.López-Cabrera ›3met al.›m,J.Chem.Phys.›1m97›m,8481(1992).

3. V.S.Popov, A.V.Sergeev, Phys.Lett.A ›1m172›m, 193, (1993).

4. A.Schmid, Ann.Phys. (N.Y.) ›1m170›m, 333 (1986).


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