to extend the methods related to perturbation theory (PT) on the region where the coupling constant (or perturbation parameter) appears to be strong. I believe that for most problems encountered it may be achieved in two stages: (1) derivation of recurrence relations between PT coefficients which makes it possible to calculate large number of them by computer; (2) selection of suitable generalized summation procedure such as Borel method or Padé approximants in order to transform divergent PT series into convergent sequence of approximants.

In my recent work, the perturbation parameter is 1/*N*, where
*N* is the dimensionality of space. My plans for the future are
related mainly to the second stage.

__Prehistory of my present research__

In my earliest papers [1, 2] the methods of PT were applied for calculating the energies and widths of bound and exited states in spherically-symmetric screened Coulomb potentials, especially Yukawa potential. We used large-order PT in powers of screening parameter.

Summation of PT series for a well-known problem of Stark effect
in a hydrogen atom was considered in [4, 6, 11]. For strong field, the
main problem is how to obtain *complex* energies by summation of
*real* terms of PT series. We used special summation procedure,
namely quadratic Padé - Hermite approximants. Convenient recurrence
relations for calculation of such approximants were derived in [3].

In collaboration with V.S.Popov and co-workers from ITEP
(Moscow), I was involved in 1/N-expansion for various quantum-
mechanical problems, such as screened Coulomb potentials, Stark and
Zeeman effects, helium-like ions, and two-center-Coulomb problem [7,
9 - 11, 13]. Somewhere we use the different term "1/*n*-expansion",
because we expand the energy in powers of 1/*n*, where *n* =
*l *+ *nr* + 1, *nr* = const, and *l***. Our approach is
equivalent to 1/*N*-expansion, because we arrive to the same radial
Schrödinger equation. I have calculated about 50 expansion coefficients
for spherically-symmetric one-particle problems and about 10
coefficients for the case of axially symmetric or three-particle problems.
Usually, Padé approximants were used to sum the series. Particularly,
we were interested in quasistationary states, when an effective potential
has no real minimum, so the 1/*N*-expansion is complex (for
example, strong-field Stark effect).

Recently we examined the asymptotics of large orders of the
coefficients in 1/*N*-expansion [18 - 21, 25]. Typically, they grow
as factorials, e(*k*) * *C*0 *ak k*b *k*! with *k* * *.
We found the parameters *C*0, *a* and b by means of
dispersion relations including an integral from the imaginary part of the
energy. Particularly , *a*-1 equals to the action integral standing in
the exponent in the quasiclassical formula for decay rate:

*r*1

*a*-1 = * [2(*V*eff(*r*)-*V*0 )]1/2 *dr*

*r*0

where *V*0=*V*eff(*r*0) is the minimum of the effective
potential, and *r*1 is a turning point, *V*eff(*r*1)=*V*0
[19]. For bound states, there is a pair of complex-conjugate turning
points, so the large-order asymptotics contains two terms: e(*k*) *
(*C*0 *ak* + *C*0*** *a*k k*b) *k*! where
*C*0 and *a* are complex constants.

__Details of my present and future work__

In my recent research, I extend the earlier results on asymptotics of large
orders of 1/*N*-expansion [18 - 20] to multidimensional effective
potentials for treating nonspherically-symmetric and three-particle
systems. My approach to the 1/*N*-expansion in large order for such
systems is guided by recent studies of 1/*N*-expansion for helium
isoelectronic sequence, see:

Goodson D. Z., López-Cabrera M. et al, J.Chem.Phys. 1992, v.97, no.11, p.8481.

This paper is concerned with calculation of the expansion
coefficients to high order (*20 to 30) and with the analysis of the
singularity structure of the energy as a function of 1/*N*. It was
shown that Padé - Borel summation incorporating results of the
singularity analysis yields highly accurate energies. However, there was
accounted for only the Coulombic pole at d=1/*N*=1. An essential
singularity at d=0 responsible for the factorial growth of the expansion
coefficients was found numerically, but its origin remains to be
revealed. I guess that it would be very useful to know exactly the
singularity of the Borel function d0=1/*a*, where *a* is the
parameter of large-order asymptotics. My object is the calculation of the
parameter *a* for various multidimensional systems.

I convert the calculation of PT in large order into barrier- penetration problem by means of well-known dispersion techniques, see for example:

Banks T., Bender C.M., Wu T.T., Phys.Rev.D 1973, v.8, no.10, p.3346.

I deal with a multidimensional quantum decay problem (two- dimensional for axially symmetric system and three-dimensional for three-particle system).

The central problem is the solution of the eikonal equation and
minimization of the classical action in order to determine the parameter
*a*. 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, see:

A.Schmid, Ann.Phys.(N.Y.), 1986, v.170, p.333.

The parameter *a* equals to the reciprocal of the action along
this trajectory. In the second (quite novel) approach, the action is
expanded as a perturbation series around the minimum of the effective
potential.

As an example that has all essential features of the general problem, I was investigating a hydrogen atom in parallel electric and magnetic fields. Just recently, two papers on this topic were prepared in collaboration with V.S.Popov [21, 25].

As a simple model, I am going to reexamine the coupled anharmonic oscillators, see:

Banks T., Bender C.M., Phys.Rev.D 1973, v.8, no.10, p.3366.

My preliminary calculations reveal some inaccuracies in Table 1 from this paper.

Also, I am going to examine a more difficult problem for helium
isoelectronic sequence. Note, that there is no decay, so only complex
stationary points of the classical action exist which determine the
parameter *d*0=*a*-1. I would be able to check approximate
results for singularities *d*s**d*0 obtained by López-Cabrera et
al in 1992. Finally, I hope to improve the convergence of Borel sums by
taking into account the singularity of the Borel function
*d*0=1/*a* (namely, by using Darboux approximants,
accounting for square-root singularities).

At the moment, I am investigating 1/*N*-expansion for near-
degenerate excited states, such as |010> and |200> states of helium. For
details, see enclosed abstracts, especially the second one. Here, I only
notice that such degeneracy is very typical for higher excited states in
multidimensional problems.

Calculate Rayleigh - Schrodinger perturbation series for the
quartic, cubic, sextic,
octic anharmonic oscillators and the
Barbanis potential (two-dimensional anharmonic oscillator) using *Mathematica* programs.

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