**SEMICLASSICAL DIMENSIONAL EXPANSION**

** FOR ATOMS IN EXTERNAL FIELDS**

A.V.Sergeev

S.I.Vavilov State Optical Institute,

Tuchkov per. 1, St-Petersburg, 199034 Russia,

e-mail sergeev@soi.spb.su

The energy of bound and quasistationary states of atoms is
expanded as a power series in 1/*D*, where *D* is the dimensionality
of coordinate space. A multidimensional continuation of the problem is
performed, and *D* is treated as a continuous parameter with an initial
problem corresponding to the physical value *D*=3. It is a semiclassical
methods, which is rather similar to the method of molecular vibration analysis.
The problem reduces to the Rayleigh - Schrodinger perturbation theory for
an anharmonic oscillator. The expansion coefficients can be calculated
exactly and to high order, using recursive relations.

Because of the divergence of the 1/*D*-expansion, the special
summation methods, taking into account the behaviour of the coefficients
at large orders, are used. Earlier, the asymptotics of large orders of
the 1/*D*-expansion has been studied both numerically [1] and analytically
[2]. Typically, the coefficients in the expansion grow as factorials*
Ek* ~ *c*0* ak kb k*! Here, two aspects of the problem involved
are concerned.

Firstly, the appropriate modification of Pade - Borel summation procedure
is proposed, designed to take into account the parameters *a* and
b of the asymptotics which can be calculated exactly. So, the singularity
of the Borel function is approximated properly. As a result, the procedure
considerably accelerates the convergence.

Secondly, the divergence of the 1/D-expansion for *excited *states
is considered. A numerical test has revealed the presence of square-root
singularities of the energy function for 1s 2s 1S state of helium [3].
Here, it is proved that such singularities originate from a crossing of
energy levels. It leads to a non-factorial growth of the expansion coefficients
Ek ~ cs Dsk k-3/2, where Ds is a dimensionality when the crossing occurs.
The parameters cs and Ds are found analytically. Suitable modification
of 1/*D*-expansion is proposed which avoids the troubles related to
strong divergence of the 1/*D*-series for near-crossing energy levels.

As a typical example, a hydrogen atom subject to parallel electric and magnetic fields is investigated in detail. The large-order dimensional perturbation theory allows one to obtain highly accurate energies and widths.

1. M.Lopez-Cabrera, D.Z.Goodson, D.R.Herschbach, J.D.Morgan. Phys.Rev.Lett.
**68**, 1992 (1992); J.Chem.Phys.** 97**, 8481 (1992).

2. V.S.Popov, A.V.Sergeev, Phys.Lett.A** 172**, 193 (1993); Pisma
v ZhETF (JETP Letters) **57**, 273 (1993).

3. D.Z.Goodson, D.K.Watson, Phys.Rev.A **48**, 2668 (1993).

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Designed by A. Sergeev.