224
5 Formulation and Implementation of the Gamow Shell Model
the ˆ
J + matrix times vector operator is negligible compared to a full ˆ
P J application.
If M = J , it is convenient to apply ˆ
J 2 − J (J + 1) to the considered Gamow
shell model vector, because it is zero only if the considered Gamow shell model
vector is coupled to J . Hence, as the application of ˆ
J ± is done only once or twice
per iteration, the previous test is very fast compared to the application of ˆ
H or ˆ
P J
operators.
5.9
Diagonalization of Very Large Gamow Shell Model
Matrices with the Density Matrix Renormalization Group
Approach
In order to be able to determine Gamow shell model eigenvalues almost exactly,
the density matrix renormalization group approach has been developed and applied
in the non-Hermitian case of the Gamow shell model [38, 39]. The density matrix
renormalization group approach [40, 41] allows to calculate eigenstates of matrices
whose dimensions are beyond capabilities of standard Krylov methods. The method
has been tested in nuclear physics in the frame of standard shell model [42–45]. It
was generalized to the non-Hermitian Gamow shell model several years ago, firstly
for a space of valence neutrons [38], and then for a for space containing both protons
and neutrons [39].
The application of density matrix renormalization group approach to Hermitian
matrices of the nuclear shell model has shown that convergence to an exact result is
difficult to attain due to the large matrix elements of inter-shell coupling of the
nucleon-nucleon interaction [43–45]. In fact, the density matrix renormalization
group is especially effective when the couplings induced by the used interactions are
weak. This is particularly the case for spin chains [40, 41], where obtained binding
energies are almost exact.
Initially, the density matrix renormalization group method had been developed in
the context of real-energy physics only [40–42]. Consequently, the density matrix
renormalization group method had firstly to be generalized to the use of nonHermitian Hamiltonians in order to applied in the context of the Gamow shell model.
While the construction of the complex symmetric matrices representing reduced
density and Hamiltonian operators in the Berggren basis is straightfoward within the
theory of rigged Hilbert spaces (see Sect. 5.1.1), the problem of the identification
of resonance eigenstates among the numerous scattering eigenstates remains, as
in the Gamow shell model (see Sect. 5.4). In order to solve this problem, one can
note that continuum coupling typically generates small configuration mixing among
scattering configurations. Hence, the use of the overlap method (see Sect. 5.4),
devised in the Gamow shell model along with the Jacobi-Davidson method (see
Sect. 5.7), can be applied as well in the density matrix renormalization group
framework. This allows to efficiently calculate the eigenstates of the Hamiltonian
arising from the density matrix renormalization group method. Moreover, due to the
weak couplings occurring between the pole and scattering part of the Gamow shell
5 Formulation and Implementation of the Gamow Shell Model
the ˆ
J + matrix times vector operator is negligible compared to a full ˆ
P J application.
If M = J , it is convenient to apply ˆ
J 2 − J (J + 1) to the considered Gamow
shell model vector, because it is zero only if the considered Gamow shell model
vector is coupled to J . Hence, as the application of ˆ
J ± is done only once or twice
per iteration, the previous test is very fast compared to the application of ˆ
H or ˆ
P J
operators.
5.9
Diagonalization of Very Large Gamow Shell Model
Matrices with the Density Matrix Renormalization Group
Approach
In order to be able to determine Gamow shell model eigenvalues almost exactly,
the density matrix renormalization group approach has been developed and applied
in the non-Hermitian case of the Gamow shell model [38, 39]. The density matrix
renormalization group approach [40, 41] allows to calculate eigenstates of matrices
whose dimensions are beyond capabilities of standard Krylov methods. The method
has been tested in nuclear physics in the frame of standard shell model [42–45]. It
was generalized to the non-Hermitian Gamow shell model several years ago, firstly
for a space of valence neutrons [38], and then for a for space containing both protons
and neutrons [39].
The application of density matrix renormalization group approach to Hermitian
matrices of the nuclear shell model has shown that convergence to an exact result is
difficult to attain due to the large matrix elements of inter-shell coupling of the
nucleon-nucleon interaction [43–45]. In fact, the density matrix renormalization
group is especially effective when the couplings induced by the used interactions are
weak. This is particularly the case for spin chains [40, 41], where obtained binding
energies are almost exact.
Initially, the density matrix renormalization group method had been developed in
the context of real-energy physics only [40–42]. Consequently, the density matrix
renormalization group method had firstly to be generalized to the use of nonHermitian Hamiltonians in order to applied in the context of the Gamow shell model.
While the construction of the complex symmetric matrices representing reduced
density and Hamiltonian operators in the Berggren basis is straightfoward within the
theory of rigged Hilbert spaces (see Sect. 5.1.1), the problem of the identification
of resonance eigenstates among the numerous scattering eigenstates remains, as
in the Gamow shell model (see Sect. 5.4). In order to solve this problem, one can
note that continuum coupling typically generates small configuration mixing among
scattering configurations. Hence, the use of the overlap method (see Sect. 5.4),
devised in the Gamow shell model along with the Jacobi-Davidson method (see
Sect. 5.7), can be applied as well in the density matrix renormalization group
framework. This allows to efficiently calculate the eigenstates of the Hamiltonian
arising from the density matrix renormalization group method. Moreover, due to the
weak couplings occurring between the pole and scattering part of the Gamow shell