9 Operator Perturbation Theory for Atomic Systems
163
problem containing both bound states and resonance (complex) EE values as solutions. They have stated the extended virial theorem and developed an original
approach to determination of the resonance eigenvalues by means of elementary
matrix manipulations. The error estimates for the approximate complex eigenvalues of the dilated Schrödinger operator are derived in Ref. [24], where the calculation data for the resonances of the DC Stark effect in the hydrogen are presented.
In the complex-coordinate method a dilation transformation is used to make the
resonance EF square integrable. The resonance of nondilation analytic potentials
can be obtained numerically by using Simons exterior-scaling procedures within
the finite-basis-set approximation [27, 28]. The exterior-scaling procedure has been
used only with direct numerical integration methods [27–30]. The use of a finite
basis set in these calculations will enable one to use numerical techniques developed for bound states in calculating resonance positions and widths for nondilation
potentials [27–36].
Rao, Liu and Li [18] have studied theoretically the DC strong-field Stark resonances by a complex-scaling plus B-spline approach and shown that the high accuracy is attributed to the good stationarity behavior of eight trajectories with a
well-adjusted 8-spline basis. Rao and Li [19] have also studied the behavior of
the resonances of a hydrogen atom in parallel magnetic and electric fields with a
complex scaling plus B-spline method too and received a consistent data on the
corresponding resonance parameters in dependence upon the ratio of the magneticfield strength to the electric-field strength. It is worth to remind that the similar
approaches have been developed to describe the Zeemane resonances. Namely, for
hydrogen atoms in pure magnetic fields, the properties of resonant states were calculated by the complex scaling, the R matrix, the operator PT (OPT) and other
methods (look, for example, [4–7]. The generalization of methods to account for
the resonance interference, non-H and relativistic effects is still an important problem, though here a definite progress has been reached too. One should mention such
approaches as a model potential method, quantum defect approximation, the OPT,
complex scaling plus B-spline method etc. [3–19, 64–76]. Regarding the quantum
chaos phenomenon in atoms in electromagnetic fields (look, for example, [77–80])
note that this topic should not be considered here. Let us only note that the approach presented below together with the various methods of the theory of chaos
in options [80–82] has been effectively used to describe the chaotic behavior of the
hydrogen and non-H atoms in the magnetic and microwave fields.
Here a consistent uniform quantum-mechanical approach to the solution of the
non-stationary state problems including the DC strong-field Stark effect and also
scattering problem is presented. It allows calculation of complex EE and especially is destined for investigation of the spectral region near the new continuum
boundary. The essence of the method is the inclusion of the well known “distorted
waves approximation” method in the frame of the formally exact PT. The zero-order
Hamiltonian H 0 of this PT possesses only stationary bound and scattering states.
To overcome formal difficulties, we define the zero-order Hamiltonian by the set
of orthogonal eigenfunctions (EF) and EE without specifying the explicit form of
the corresponding zeroth-order potential. To ensure rapid PT convergence, a physically reasonable spectrum (EE and EF) must be chosen as the zero order, similar
163
problem containing both bound states and resonance (complex) EE values as solutions. They have stated the extended virial theorem and developed an original
approach to determination of the resonance eigenvalues by means of elementary
matrix manipulations. The error estimates for the approximate complex eigenvalues of the dilated Schrödinger operator are derived in Ref. [24], where the calculation data for the resonances of the DC Stark effect in the hydrogen are presented.
In the complex-coordinate method a dilation transformation is used to make the
resonance EF square integrable. The resonance of nondilation analytic potentials
can be obtained numerically by using Simons exterior-scaling procedures within
the finite-basis-set approximation [27, 28]. The exterior-scaling procedure has been
used only with direct numerical integration methods [27–30]. The use of a finite
basis set in these calculations will enable one to use numerical techniques developed for bound states in calculating resonance positions and widths for nondilation
potentials [27–36].
Rao, Liu and Li [18] have studied theoretically the DC strong-field Stark resonances by a complex-scaling plus B-spline approach and shown that the high accuracy is attributed to the good stationarity behavior of eight trajectories with a
well-adjusted 8-spline basis. Rao and Li [19] have also studied the behavior of
the resonances of a hydrogen atom in parallel magnetic and electric fields with a
complex scaling plus B-spline method too and received a consistent data on the
corresponding resonance parameters in dependence upon the ratio of the magneticfield strength to the electric-field strength. It is worth to remind that the similar
approaches have been developed to describe the Zeemane resonances. Namely, for
hydrogen atoms in pure magnetic fields, the properties of resonant states were calculated by the complex scaling, the R matrix, the operator PT (OPT) and other
methods (look, for example, [4–7]. The generalization of methods to account for
the resonance interference, non-H and relativistic effects is still an important problem, though here a definite progress has been reached too. One should mention such
approaches as a model potential method, quantum defect approximation, the OPT,
complex scaling plus B-spline method etc. [3–19, 64–76]. Regarding the quantum
chaos phenomenon in atoms in electromagnetic fields (look, for example, [77–80])
note that this topic should not be considered here. Let us only note that the approach presented below together with the various methods of the theory of chaos
in options [80–82] has been effectively used to describe the chaotic behavior of the
hydrogen and non-H atoms in the magnetic and microwave fields.
Here a consistent uniform quantum-mechanical approach to the solution of the
non-stationary state problems including the DC strong-field Stark effect and also
scattering problem is presented. It allows calculation of complex EE and especially is destined for investigation of the spectral region near the new continuum
boundary. The essence of the method is the inclusion of the well known “distorted
waves approximation” method in the frame of the formally exact PT. The zero-order
Hamiltonian H 0 of this PT possesses only stationary bound and scattering states.
To overcome formal difficulties, we define the zero-order Hamiltonian by the set
of orthogonal eigenfunctions (EF) and EE without specifying the explicit form of
the corresponding zeroth-order potential. To ensure rapid PT convergence, a physically reasonable spectrum (EE and EF) must be chosen as the zero order, similar