6.4 Three-Body Model in Berggren Basis
269
Besides this different interpretation of basis wave functions, the derivation of the
Berggren basis using the hyper-radius ρ instead of radius r is formally identical.
Indeed, the Coulomb potential defining the wave functions depending on the hyperradius ρ is formally the same as that used in Sect. 3.5. However, in order to
emphasize the different physical content of calculated wave functions, one will use
in this section the notation B(k, ρ) instead of u(k, r). Let us note that the Berggren
basis developed for particle states can also be generalized to pairs of nucleons. This
will be the subject of Sect. 6.5, where the extension of Richardson pairing model to
the continuum will be realized in the Berggren ensemble.
Similarly to Eq. (3.66), the Berggren completeness relation using the hyperradius ρ reads as:
n∈b,d
B n (k n , ρ)B n (k n , ρ
) +
L +
B(k, ρ)B(k, ρ
)dk = δ(ρ − ρ
) ,
(6.24)
where b and d stand for bound and decaying states, respectively. For numerical
purposes, the L + contour has to be discretized, e.g. by adopting the Gauss-Legendre
quadrature [105]. Due to the symmetry between the second and fourth quadrants
in complex momentum plane, only the contour in the fourth quadrant L + needs
to be considered. If the contour L + is chosen along the real k-axis, the Berggren
completeness relation reduces to the Newton completeness relation involving bound
and real-energy scattering states.
The total wave function, whose quantum numbers are total angular momentum
J , total angular momentum projection M, and parity π, can be written as a linear
combination of basis states of coordinates Ω 5 and ρ:
Ψ
J Mπ (ρ, Ω 5 ) = ρ
−5/2
γ K
ψ
J π
γ K (ρ)Y
J M
γ K (Ω 5 ) ,
(6.25)
where γ = {s 1 , s 2 , s 3 , S 12 , S, , x , , y , L} is a set of quantum numbers. s and l stand
for spin and orbital angular momentum, respectively, ψ J π
γ K (ρ) is the hyperradial
wavefunction, and the hyperangular basis state Y J M
γ K (Ω 5 ) is the hyperspherical
harmonics [106].
In order to consider the continuum and resonances precisely, one uses the
Berggren basis expansion for the hyperradial wavefunction:
ψ
J π
γ K (ρ) =
n
C
J πM
γ nK B
J π
γ n (ρ) ,
(6.26)
where B J π
γ n (ρ) is the Berggren basis and C
J πM
γ nK is the expansion coefficient. As
a result, the hyperradial Schrödinger equation can be written as a set of coupled-
269
Besides this different interpretation of basis wave functions, the derivation of the
Berggren basis using the hyper-radius ρ instead of radius r is formally identical.
Indeed, the Coulomb potential defining the wave functions depending on the hyperradius ρ is formally the same as that used in Sect. 3.5. However, in order to
emphasize the different physical content of calculated wave functions, one will use
in this section the notation B(k, ρ) instead of u(k, r). Let us note that the Berggren
basis developed for particle states can also be generalized to pairs of nucleons. This
will be the subject of Sect. 6.5, where the extension of Richardson pairing model to
the continuum will be realized in the Berggren ensemble.
Similarly to Eq. (3.66), the Berggren completeness relation using the hyperradius ρ reads as:
n∈b,d
B n (k n , ρ)B n (k n , ρ
) +
L +
B(k, ρ)B(k, ρ
)dk = δ(ρ − ρ
) ,
(6.24)
where b and d stand for bound and decaying states, respectively. For numerical
purposes, the L + contour has to be discretized, e.g. by adopting the Gauss-Legendre
quadrature [105]. Due to the symmetry between the second and fourth quadrants
in complex momentum plane, only the contour in the fourth quadrant L + needs
to be considered. If the contour L + is chosen along the real k-axis, the Berggren
completeness relation reduces to the Newton completeness relation involving bound
and real-energy scattering states.
The total wave function, whose quantum numbers are total angular momentum
J , total angular momentum projection M, and parity π, can be written as a linear
combination of basis states of coordinates Ω 5 and ρ:
Ψ
J Mπ (ρ, Ω 5 ) = ρ
−5/2
γ K
ψ
J π
γ K (ρ)Y
J M
γ K (Ω 5 ) ,
(6.25)
where γ = {s 1 , s 2 , s 3 , S 12 , S, , x , , y , L} is a set of quantum numbers. s and l stand
for spin and orbital angular momentum, respectively, ψ J π
γ K (ρ) is the hyperradial
wavefunction, and the hyperangular basis state Y J M
γ K (Ω 5 ) is the hyperspherical
harmonics [106].
In order to consider the continuum and resonances precisely, one uses the
Berggren basis expansion for the hyperradial wavefunction:
ψ
J π
γ K (ρ) =
n
C
J πM
γ nK B
J π
γ n (ρ) ,
(6.26)
where B J π
γ n (ρ) is the Berggren basis and C
J πM
γ nK is the expansion coefficient. As
a result, the hyperradial Schrödinger equation can be written as a set of coupled-