SEMICLASSICAL DIMENSIONAL EXPANSION
FOR ATOMS IN EXTERNAL FIELDS
S.I.Vavilov State Optical Institute,
Tuchkov per. 1, St-Petersburg, 199034 Russia,
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  and analytically . Typically, the coefficients in the expansion grow as factorials Ek ~ c0 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 . 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.
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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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