160
4 Two-Particle Systems in the Berggren Basis
where E J is the energy of the system and
V
J
cc (r) =
λ
J
c j c |P λ (cos θ)|Θ
J
c j c
V λ (r)
(4.20)
is the channel-channel coupling potential.
The standard method to solve Eq. (4.19) is to use the direct integration of
coupled-channel equations. While it is apparently the most precise method in
this case, it becomes unstable when the number of channels is large. Moreover,
the presence of coupling potentials in the dipolar case, whose radial parts are
proportional to 1/r 2 for large r, prevents from having analytical asymptotes of
channel wave functions. This is a problem for the direct integration as in this method
one needs to impose asymptotic boundary conditions from an exact outgoing wave
function which are unavailable hereby. However, direct integration can be used
conveniently if the moment of inertia is infinite. In this case, as will be shown in
Exercises IV and V, Eq. (4.19) possesses analytical solutions for large r, so that
coupled-channel wave functions can be exactly matched to their asymptotic form
and hence, the Berggren expansion method can be numerically tested against the
direct integration method.
The Berggren expansion method for solving the particle-rotor model has been
introduced in Ref. [44]. In this method, the Hamiltonian is diagonalized in a complete Berggren ensemble of single-particle states [45–47] which is generated by a
finite-depth spherical one-body potential. While the finite-depth potential generating
the Berggren ensemble can be chosen arbitrarily, to improve the convergence, one
takes the diagonal part of the channel coupling potential V cc (r). The basis states
Φ k,c (r) are eigenstates of the spherical potential V cc (r). These states are regular
at the origin and meet outgoing boundary conditions for bound states (b), decaying
states (d), or scattering states (s). Note that the wave number k of eigenstates Φ k,c (r)
is in general complex. The normalization of the bound states is standard, while
that for the decaying states involves the exterior complex scaling [44, 48, 49]. The
scattering states are normalized to the Dirac delta function.
To determine Berggren ensemble, one calculates for all chosen partial waves
first the single-particle bound and resonance states of the basis-generating potential.
Then, for each channel ((, j ), one selects the contour L
+
,j in a fourth quadrant
of the complex k-plane. All ((, j )-scattering states in this ensemble belong to the
preselected contour L
+
,j . The set of all resonant states and all scattering states
on the contour L
+
c ,j c
form a complete single-particle basis for the channel ((, j ).
Each contour L
+
,j is composed of three segments: the first one from the origin to
k peak in the fourth quadrant of the complex k-plane, the second one from k peak to
k middle on the real k-axis (R(k) > 0), and the third one from k middle to k max also on
the real k-axis. The segments are discretized with the Gauss-Legendre quadrature,
which has been seen to be most precise numerically (see Sect. 3.7.1). The Berggren
4 Two-Particle Systems in the Berggren Basis
where E J is the energy of the system and
V
J
cc (r) =
λ
J
c j c |P λ (cos θ)|Θ
J
c j c
V λ (r)
(4.20)
is the channel-channel coupling potential.
The standard method to solve Eq. (4.19) is to use the direct integration of
coupled-channel equations. While it is apparently the most precise method in
this case, it becomes unstable when the number of channels is large. Moreover,
the presence of coupling potentials in the dipolar case, whose radial parts are
proportional to 1/r 2 for large r, prevents from having analytical asymptotes of
channel wave functions. This is a problem for the direct integration as in this method
one needs to impose asymptotic boundary conditions from an exact outgoing wave
function which are unavailable hereby. However, direct integration can be used
conveniently if the moment of inertia is infinite. In this case, as will be shown in
Exercises IV and V, Eq. (4.19) possesses analytical solutions for large r, so that
coupled-channel wave functions can be exactly matched to their asymptotic form
and hence, the Berggren expansion method can be numerically tested against the
direct integration method.
The Berggren expansion method for solving the particle-rotor model has been
introduced in Ref. [44]. In this method, the Hamiltonian is diagonalized in a complete Berggren ensemble of single-particle states [45–47] which is generated by a
finite-depth spherical one-body potential. While the finite-depth potential generating
the Berggren ensemble can be chosen arbitrarily, to improve the convergence, one
takes the diagonal part of the channel coupling potential V cc (r). The basis states
Φ k,c (r) are eigenstates of the spherical potential V cc (r). These states are regular
at the origin and meet outgoing boundary conditions for bound states (b), decaying
states (d), or scattering states (s). Note that the wave number k of eigenstates Φ k,c (r)
is in general complex. The normalization of the bound states is standard, while
that for the decaying states involves the exterior complex scaling [44, 48, 49]. The
scattering states are normalized to the Dirac delta function.
To determine Berggren ensemble, one calculates for all chosen partial waves
first the single-particle bound and resonance states of the basis-generating potential.
Then, for each channel ((, j ), one selects the contour L
+
,j in a fourth quadrant
of the complex k-plane. All ((, j )-scattering states in this ensemble belong to the
preselected contour L
+
,j . The set of all resonant states and all scattering states
on the contour L
+
c ,j c
form a complete single-particle basis for the channel ((, j ).
Each contour L
+
,j is composed of three segments: the first one from the origin to
k peak in the fourth quadrant of the complex k-plane, the second one from k peak to
k middle on the real k-axis (R(k) > 0), and the third one from k middle to k max also on
the real k-axis. The segments are discretized with the Gauss-Legendre quadrature,
which has been seen to be most precise numerically (see Sect. 3.7.1). The Berggren