1mLarge-order dimensional expansion for three-body systemsm
3mS.I.Vavilov State Optical Institute,
Tuchkov per. 1, Saint Petersburg, 199034 Russian Federation,
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 . 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 . 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, . For bound states, there
is a pair of complex-conjugate turning points, so the large-order
asymptotics contains two terms.
Here the earlier results  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
. 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
1. A.V.Sergeev,Yadernaya Fizika 1m50m,945(1989)[Sov.J.of Nucl.Phys].
2. D.Z.Goodson, M.López-Cabrera 3met al.m,J.Chem.Phys.1m97m,8481(1992).
3. V.S.Popov, A.V.Sergeev, Phys.Lett.A 1m172m, 193, (1993).
4. A.Schmid, Ann.Phys. (N.Y.) 1m170m, 333 (1986).
Back to Presentations on conferences.
Designed by A. Sergeev.