162
A.V. Glushkov
vironments etc. [4–14]. Naturally in the last two decades a great progress has been
made on the Stark effect for the hydrogen atom as well as for non-H atoms [2–62].
An external electric field shifts and broadens the bound state atomic levels.
The standard quantum -mechanical approach relates complex eigenenergies (EE)
E = E r + iΓ /2 and complex eigenfunctions (EF) to the shape resonances. The field
effects drastically increase upon going from one excited level to another. The highest
levels overlap forming a “new continuum” with lowered boundary. The calculation
difficulties inherent to the standard quantum mechanical approach are well known.
Here one should mention the well-known Dyson phenomenon. The WentzelKramers-Brillouin (WKB) approximation overcomes these difficulties for the states
lying far from the “new continuum” boundary. Some modifications of the WKB
method [4, 6–8, 50, 56, 63–66] are introduced in [4, 50, 66] and Glushkov, Ivanov
and Letokhov, where the first theoretical estimation of the effectiveness of the selective ionization of the Rydberg atom using electric and laser fields has been fulfilled.
The usual WKB approximation applicability is substantiated in the case of a relatively weak electric field [2, 3]. One can show that the standard form of the WKB
method applicability condition can be reformulated as the requirement that the examined resonances be well separated one from other. The same is so regarding the
widespread asymptotic phase method [42], based on the Breit-Wigner parameterization for the asymptotic phase shift dependence on scattering energy and the method
by Luc-Koenig and Bachelier, who have used a normalization constant [42, 48]. Different calculational procedures are used in the Pade and then Borel summation of the
divergent Rayleigh-Schrödinger perturbation theory (PT) series [45, 66] and in the
sufficiently exact numerical solution of the difference equations following from expansion of the wave function over finite basis [41, 46, 48, 51, 52], complex-scaling
method [17–55]. It should be noted that the latter has been extensively used to describe the resonance behavior in different atomic and even molecular systems. Its
mathematical foundation is linked with the theory of dilatation analyticity [27, 28].
Surely, though the Hamiltonian of an atom in a DC electric field is not a dilatational
analytic operator, Reinhardt [44] has performed the numerical experiments on the
diagonalization of the complex-scaled Stark Hamiltonian for a hydrogen with a real
L basis set. The same method has been used by Cerjan et al. [40] to get new data on
the ground and low-excited states of a hydrogen atom in a DC and AC fields. Farrelly
and Reinhardt [47] have used the complex coordinate rotation method in combination with numerical integration of the separated equation. Ivanov and Ho [54] have
applied the method for the Dirac Hamiltonian. Different applications are reviewed
in Ref. [53].
Hehenberger, McIntosh and E. Brändas [21] have applied the Weyl’s theory to
the Stark effect in the hydrogen atom. They have shown that one of the interesting features of Weyl’s theory is that it requires a complex parameter and complex
solutions to the differential equations making it a powerful tool for the treatment
of resonance states [21]. Rittby, Elander and Brändas [25] have applied the Weyl’s
theory and the complex-rotation method to phenomena associated with a continuum
spectrum. Brändas and Froelich [23] have shown that a complex scale transformation of the time-dependent Schrödinger equation leads to a symmetric EE value
A.V. Glushkov
vironments etc. [4–14]. Naturally in the last two decades a great progress has been
made on the Stark effect for the hydrogen atom as well as for non-H atoms [2–62].
An external electric field shifts and broadens the bound state atomic levels.
The standard quantum -mechanical approach relates complex eigenenergies (EE)
E = E r + iΓ /2 and complex eigenfunctions (EF) to the shape resonances. The field
effects drastically increase upon going from one excited level to another. The highest
levels overlap forming a “new continuum” with lowered boundary. The calculation
difficulties inherent to the standard quantum mechanical approach are well known.
Here one should mention the well-known Dyson phenomenon. The WentzelKramers-Brillouin (WKB) approximation overcomes these difficulties for the states
lying far from the “new continuum” boundary. Some modifications of the WKB
method [4, 6–8, 50, 56, 63–66] are introduced in [4, 50, 66] and Glushkov, Ivanov
and Letokhov, where the first theoretical estimation of the effectiveness of the selective ionization of the Rydberg atom using electric and laser fields has been fulfilled.
The usual WKB approximation applicability is substantiated in the case of a relatively weak electric field [2, 3]. One can show that the standard form of the WKB
method applicability condition can be reformulated as the requirement that the examined resonances be well separated one from other. The same is so regarding the
widespread asymptotic phase method [42], based on the Breit-Wigner parameterization for the asymptotic phase shift dependence on scattering energy and the method
by Luc-Koenig and Bachelier, who have used a normalization constant [42, 48]. Different calculational procedures are used in the Pade and then Borel summation of the
divergent Rayleigh-Schrödinger perturbation theory (PT) series [45, 66] and in the
sufficiently exact numerical solution of the difference equations following from expansion of the wave function over finite basis [41, 46, 48, 51, 52], complex-scaling
method [17–55]. It should be noted that the latter has been extensively used to describe the resonance behavior in different atomic and even molecular systems. Its
mathematical foundation is linked with the theory of dilatation analyticity [27, 28].
Surely, though the Hamiltonian of an atom in a DC electric field is not a dilatational
analytic operator, Reinhardt [44] has performed the numerical experiments on the
diagonalization of the complex-scaled Stark Hamiltonian for a hydrogen with a real
L basis set. The same method has been used by Cerjan et al. [40] to get new data on
the ground and low-excited states of a hydrogen atom in a DC and AC fields. Farrelly
and Reinhardt [47] have used the complex coordinate rotation method in combination with numerical integration of the separated equation. Ivanov and Ho [54] have
applied the method for the Dirac Hamiltonian. Different applications are reviewed
in Ref. [53].
Hehenberger, McIntosh and E. Brändas [21] have applied the Weyl’s theory to
the Stark effect in the hydrogen atom. They have shown that one of the interesting features of Weyl’s theory is that it requires a complex parameter and complex
solutions to the differential equations making it a powerful tool for the treatment
of resonance states [21]. Rittby, Elander and Brändas [25] have applied the Weyl’s
theory and the complex-rotation method to phenomena associated with a continuum
spectrum. Brändas and Froelich [23] have shown that a complex scale transformation of the time-dependent Schrödinger equation leads to a symmetric EE value