5.4 Determination of Eigenvalues in Gamow Shell Model: The Overlap Method
205
5.4
Determination of Eigenvalues in Gamow Shell Model:
The Overlap Method
Nuclear states in Gamow shell model are determined by the diagonalization of the
Hamiltonian matrix expressed in a basis of Slater determinants, in a way formally
identical to standard shell model. The fundamental difference is that the Slater
determinants in Gamow shell model are built from the one-body states of the
discretized completeness relation of Eq. (3.82). This implies that all the matrix
elements of the Gamow shell model Hamiltonian must be calculated by direct
integration, utilizing explicitly the radial wave functions of the Berggren basis states.
The representation of the Gamow shell model Hamiltonian in a Berggren basis of
discretized states is a complex symmetric matrix, as the basis states bear a complex
energy.
In order to diagonalize the standard shell model matrices, the Lanczos method
is widely employed, as it allows to determine low-energy nuclear states without
having to fully diagonalize the Hamiltonian matrix. However, Lanczos method
cannot be applied directly in Gamow shell model, because resonance A-body states
are surrounded by many scattering A-body states. Hence, the low-energy spectrum
obtained by the Lanczos method in Gamow shell model possesses bound, resonance,
and scattering A-body states, where resonance and scattering states cannot be
separated one from another. It is thus impossible to know if a Gamow shell model
eigenstate is a resonance or scattering state considering only its eigenenergy. In fact,
only bound states can be identified unambiguously, as they always bear negative
eigenenergies.
The solution to this problem is the use of the overlap method [21]. It is a two-step
method:
• The Hamiltonian is firstly diagonalized at the level of pole approximation, where
the basis of Slater determinants is generated only by bound and resonance onebody states of the Berggren basis. The |Ψ 0 eigenvector, which is obtained from a
full diagonalization of the Hamiltonian matrix at the level of pole approximation,
is the zeroth-order approximation of the exact eigenvector.
• The Hamiltonian is then diagonalized by taking into account all one-body states
of the Berggren basis, bound, resonance, and scattering, for which |Ψ 0 is the
initial pivot. The Gamow shell model eigenstate |Ψ corresponding to the sought
bound or resonance A-body state is the one maximizing the | |Ψ 0 |Ψ | overlap.
Let us consider the example of 20 O, where four valence neutrons interact above
a 16 O core, modeled by a Woods-Saxon potential. The shell model space consists
of the 0d 5/2 and 1s 1/2 bound states, 0d 3/2 resonance of the Woods-Saxon potential,
and of the nonresonant continuum of d 3/2 complex-energy scattering states. The
eigenstates determined by the overlap method are enclosed by squares (see Fig. 5.1).
Indeed, resonances should be stable with respect to the changes of the contour.
As the Berggren sets involving scattering states on slightly modified contours are
complete, the physical resonance states expanded using these different Berggren
205
5.4
Determination of Eigenvalues in Gamow Shell Model:
The Overlap Method
Nuclear states in Gamow shell model are determined by the diagonalization of the
Hamiltonian matrix expressed in a basis of Slater determinants, in a way formally
identical to standard shell model. The fundamental difference is that the Slater
determinants in Gamow shell model are built from the one-body states of the
discretized completeness relation of Eq. (3.82). This implies that all the matrix
elements of the Gamow shell model Hamiltonian must be calculated by direct
integration, utilizing explicitly the radial wave functions of the Berggren basis states.
The representation of the Gamow shell model Hamiltonian in a Berggren basis of
discretized states is a complex symmetric matrix, as the basis states bear a complex
energy.
In order to diagonalize the standard shell model matrices, the Lanczos method
is widely employed, as it allows to determine low-energy nuclear states without
having to fully diagonalize the Hamiltonian matrix. However, Lanczos method
cannot be applied directly in Gamow shell model, because resonance A-body states
are surrounded by many scattering A-body states. Hence, the low-energy spectrum
obtained by the Lanczos method in Gamow shell model possesses bound, resonance,
and scattering A-body states, where resonance and scattering states cannot be
separated one from another. It is thus impossible to know if a Gamow shell model
eigenstate is a resonance or scattering state considering only its eigenenergy. In fact,
only bound states can be identified unambiguously, as they always bear negative
eigenenergies.
The solution to this problem is the use of the overlap method [21]. It is a two-step
method:
• The Hamiltonian is firstly diagonalized at the level of pole approximation, where
the basis of Slater determinants is generated only by bound and resonance onebody states of the Berggren basis. The |Ψ 0 eigenvector, which is obtained from a
full diagonalization of the Hamiltonian matrix at the level of pole approximation,
is the zeroth-order approximation of the exact eigenvector.
• The Hamiltonian is then diagonalized by taking into account all one-body states
of the Berggren basis, bound, resonance, and scattering, for which |Ψ 0 is the
initial pivot. The Gamow shell model eigenstate |Ψ corresponding to the sought
bound or resonance A-body state is the one maximizing the | |Ψ 0 |Ψ | overlap.
Let us consider the example of 20 O, where four valence neutrons interact above
a 16 O core, modeled by a Woods-Saxon potential. The shell model space consists
of the 0d 5/2 and 1s 1/2 bound states, 0d 3/2 resonance of the Woods-Saxon potential,
and of the nonresonant continuum of d 3/2 complex-energy scattering states. The
eigenstates determined by the overlap method are enclosed by squares (see Fig. 5.1).
Indeed, resonances should be stable with respect to the changes of the contour.
As the Berggren sets involving scattering states on slightly modified contours are
complete, the physical resonance states expanded using these different Berggren