**FERMI-LIKE RESONANCES IN A RYDBERG ATOM**

A.V.Sergeev

S.I.Vavilov State Optical Institute

Tuchkov per. 1, St.Petersburg, 199034 Russia

e-mail sergeev@soi.spb.su

In a "circular" Rydberg state with large magnetic
quantum number *m* and small value of *n*- |*m*|, an electron
performs small oscillations about fixed position (r0,*z*0) corresponding
to the minimum of an effective potential *Veff*(r,*z*)=(*m*2-
1/4)/2r2+*V*(r,*z*). Its energy is

*E*(*n*1,*n*2)=*V*(r0,*z*0)+(*n*1+1/2)w1+(<
I>n2+1/2)*2+(anharmonicterms).

Here, we investigate the case when w1/w2 is a rational number
and so, two or more excited states with different (*n*1,*n*2)
may have approximately the same energy. In a molecular vibration theory,
the similar near-degenerate states are known as Fermi resonances* *(e.g.,
in CO2 molecule, where w1* 2w2).

The motivation for our study stems from large-dimensional approach.
Within its framework, the quantum- mechanical problem reduces to a classical
electrostatic problem and a subsequent vibrational analysis. Earlier, it
was shown [1] that while the lowest orders of 1/*D*-expansion yield
qualitatively correct results for a ground state of helium, it is no more
the case for an excited 1s2s1S state where the 1/*D*-series strikingly
diverges. Here, we explain the origin of the phenomenon discussed in the
paper [1]. We prove that 1s2s1S state (*n*1=0, *n*2=1, *n*3=0)
is strongly mixed with (200) state having near the same energy, and the
energy levels of two states represent the branches of a single multivalued
analytical function with a branching point at 1/*D*=-0.0114. This
near-degenerate state is highly sensitive to a perturbation caused by anharmonic
terms in a potential, and so the 1/*D*-expansion diverges. Suitable
modification of the method is proposed that avoids the troubles related
to energy quasi-crossing.

Apart from the helium problem, we study in detail a hydrogen atom in
a uniform magnetic field (circular states) as a simple instructive example.
The relation w1/w2=*k*1/*k*2**R* holds, when the field reaches
the strength **H**[*k*1/*k*2]=(*R*2- 1)1/2(*R*2+3)3/2/8*m*3,
in atomic units. Particularly, **H**[1/2]*4.010*m*-3. The branching
points connecting (10) and (02) levels are located closely to the real
axis, at **H**=**H**[2/1]**ic*-1/2*m*-7/2, where *c*=21*71/2.
By means of 1/*n*-expansion [2] we calculate the energy spectrum and
investigate the level quasi-crossings.

__References__

[1] D.Z.Goodson, D.K.Watson, Phys.Rev.A __48__, 2668 (1993).

[2] V.M.Vainberg, V.S.Popov, A.V.Sergeev, JETP __71__, 470 (1990).

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Designed by A. Sergeev.