9.2 Generalization to Many-Nucleon Projectiles
435
one has k i k max ∀i ∈ {a, b, c, d} in the two-body matrix elements entering the
target Hamiltonian. Therefore, the antisymmetry between projectile and target wave
functions for large K CM can be suppressed, and methods used in Sect. 9.1.1 can also
be utilized for many-nucleon projectiles in the frame of cluster approximation.
9.2.6.5 Generalization to Complex-Energy States
The Berggren basis possesses complex-energy states, so that it might seem that
Eq. (9.96) has to be extended to this case. However, realistic interactions cannot
be directly analytically continued to the complex plane [57]. Indeed, it has been
seen that the use of a harmonic oscillator expansion of two-body matrix elements
is necessary to make the realistic interactions suitable for applications using the
Berggren basis (see Sect. 5.6). Nevertheless, one cannot derive an equivalent of
Eq. (9.101) with harmonic oscillator states. This is due to the center-of-mass part
in Eq. (9.63) because Eq. (9.101) relating the linear momenta of one-body states
and projectile center-of-mass states is valid only for plane waves. As the plane wave
expansion of Bessel functions is done at fixed energy, the K CM -dependent relations
derived for plane waves can also be used with Bessel functions, and thus with
real-energy Berggren basis functions (Eq. (9.106)), but not with harmonic oscillator
states.
A solution to this conundrum can be found by using the analytic properties
of the wave functions of the solutions of the coupled-channel equations of the
Gamow shell model. For this, one firstly derives the coupled-channel equations of
the Gamow shell model within the framework of resonating group method with real
energies, so that Eq. (9.96) is valid. Then, once the coupled-channel equations of the
Gamow shell model are obtained on the real-energy axis, the real-energy solutions
of the coupled-channel equations can be analytically continued to the complexenergy plane. Consequently, one can use the coupled-channel Hamiltonian of the
resonating group method with complex-energy states, even though the coupledchannel equations were derived using real energies.
9.2.7 Coupled-Channel Equations and Wave Functions
for Structure and Reactions with Multi-Nucleon Projectiles
Let us now derive the representation of Hamiltonian in coupled-channel equations
for multi-nucleon projectiles. The procedure we follow in this derivation is similar to
the one-nucleon case. The multi-nucleon composite is expressed in a basis of Slater
determinants in order to be able to fulfill the antisymmetry requirement. Then, by
separating the Hamiltonian in low- and high-energy parts, one is able to determine
the coupling potentials entering the coupled-channel equations of the Gamow shell
model.
435
one has k i k max ∀i ∈ {a, b, c, d} in the two-body matrix elements entering the
target Hamiltonian. Therefore, the antisymmetry between projectile and target wave
functions for large K CM can be suppressed, and methods used in Sect. 9.1.1 can also
be utilized for many-nucleon projectiles in the frame of cluster approximation.
9.2.6.5 Generalization to Complex-Energy States
The Berggren basis possesses complex-energy states, so that it might seem that
Eq. (9.96) has to be extended to this case. However, realistic interactions cannot
be directly analytically continued to the complex plane [57]. Indeed, it has been
seen that the use of a harmonic oscillator expansion of two-body matrix elements
is necessary to make the realistic interactions suitable for applications using the
Berggren basis (see Sect. 5.6). Nevertheless, one cannot derive an equivalent of
Eq. (9.101) with harmonic oscillator states. This is due to the center-of-mass part
in Eq. (9.63) because Eq. (9.101) relating the linear momenta of one-body states
and projectile center-of-mass states is valid only for plane waves. As the plane wave
expansion of Bessel functions is done at fixed energy, the K CM -dependent relations
derived for plane waves can also be used with Bessel functions, and thus with
real-energy Berggren basis functions (Eq. (9.106)), but not with harmonic oscillator
states.
A solution to this conundrum can be found by using the analytic properties
of the wave functions of the solutions of the coupled-channel equations of the
Gamow shell model. For this, one firstly derives the coupled-channel equations of
the Gamow shell model within the framework of resonating group method with real
energies, so that Eq. (9.96) is valid. Then, once the coupled-channel equations of the
Gamow shell model are obtained on the real-energy axis, the real-energy solutions
of the coupled-channel equations can be analytically continued to the complexenergy plane. Consequently, one can use the coupled-channel Hamiltonian of the
resonating group method with complex-energy states, even though the coupledchannel equations were derived using real energies.
9.2.7 Coupled-Channel Equations and Wave Functions
for Structure and Reactions with Multi-Nucleon Projectiles
Let us now derive the representation of Hamiltonian in coupled-channel equations
for multi-nucleon projectiles. The procedure we follow in this derivation is similar to
the one-nucleon case. The multi-nucleon composite is expressed in a basis of Slater
determinants in order to be able to fulfill the antisymmetry requirement. Then, by
separating the Hamiltonian in low- and high-energy parts, one is able to determine
the coupling potentials entering the coupled-channel equations of the Gamow shell
model.