Although the complex scaling method clearly works with any kind of particles in
a finite system, here we explicitly assume that we are dealing with electrons
temporarily trapped by simple potentials representing atoms or molecules. The
electrons eventually tunnel out to a far-away region which we do not wish to
represent explicitly in the calculation. We are thus dealing with the specific case
of an open quantum system where we only have particles leaving the system.
3 Calculation of Resonances from Complex Scaling
The complex scaling method was initially developed by Aguilar, Balslev, Combes,
and Simon [19–21], and is based on a scaling r ! re
iθ of the position variable in the
Schr€ odinger equation. This is referred to as uniform complex scaling. Here we
review uniform complex scaling in the simple case of independent particles. Most
recent work is based on a later generalization called exterior complex scaling [22],
which we consider later. The following is a rather informal description of complex
scaling, focusing on a few important cases. More information can be found in any of
the many existing reviews.[23–27]
3.1 Formalism
Consider the standard independent-particle time-independent Schr€ odinger equation
for a finite system:
^
H ψ r
ð Þ ¼ εψ r
ð Þ:
ð3Þ
The Hamiltonian is ^
H ¼ À
1
2 ∇
2
þ v r
ð Þ, where ν(r) is some reasonably well-behaved
potential which approaches zero as r ! 1. Formally, the potential has to be
dilation or dilatation analytic [19], but the method has been applied successfully
to potentials that are not, an example of which is the Stark effect [28–30]. For our
informal review we only insist that it be analytic in relevant parts of the complex
plane.
The spectrum of H ˆ consists of a negative point spectrum corresponding to the
bound states, and the continuum ε ! 0. The goal of complex scaling is to identify
resonances associated with positive energies somewhere within the continuum.
The complex scaling operation is implemented by the operator ^
R θ defined by
^
R θ ψ r
ð Þ ¼ e
iNθ=2
ψ re
iθ
À Á ;
ð4Þ
where N is the number of spatial coordinates on which the scaling is applied (thrice
the number of particles in the 3D many-body case). θ, the scaling angle, is a fixed
Dynamical Processes in Open Quantum Systems from a TDDFT Perspective:. . .
225
a finite system, here we explicitly assume that we are dealing with electrons
temporarily trapped by simple potentials representing atoms or molecules. The
electrons eventually tunnel out to a far-away region which we do not wish to
represent explicitly in the calculation. We are thus dealing with the specific case
of an open quantum system where we only have particles leaving the system.
3 Calculation of Resonances from Complex Scaling
The complex scaling method was initially developed by Aguilar, Balslev, Combes,
and Simon [19–21], and is based on a scaling r ! re
iθ of the position variable in the
Schr€ odinger equation. This is referred to as uniform complex scaling. Here we
review uniform complex scaling in the simple case of independent particles. Most
recent work is based on a later generalization called exterior complex scaling [22],
which we consider later. The following is a rather informal description of complex
scaling, focusing on a few important cases. More information can be found in any of
the many existing reviews.[23–27]
3.1 Formalism
Consider the standard independent-particle time-independent Schr€ odinger equation
for a finite system:
^
H ψ r
ð Þ ¼ εψ r
ð Þ:
ð3Þ
The Hamiltonian is ^
H ¼ À
1
2 ∇
2
þ v r
ð Þ, where ν(r) is some reasonably well-behaved
potential which approaches zero as r ! 1. Formally, the potential has to be
dilation or dilatation analytic [19], but the method has been applied successfully
to potentials that are not, an example of which is the Stark effect [28–30]. For our
informal review we only insist that it be analytic in relevant parts of the complex
plane.
The spectrum of H ˆ consists of a negative point spectrum corresponding to the
bound states, and the continuum ε ! 0. The goal of complex scaling is to identify
resonances associated with positive energies somewhere within the continuum.
The complex scaling operation is implemented by the operator ^
R θ defined by
^
R θ ψ r
ð Þ ¼ e
iNθ=2
ψ re
iθ
À Á ;
ð4Þ
where N is the number of spatial coordinates on which the scaling is applied (thrice
the number of particles in the 3D many-body case). θ, the scaling angle, is a fixed
Dynamical Processes in Open Quantum Systems from a TDDFT Perspective:. . .
225