10
1 Introduction: From Bound States to the Continuum
Chapter 5 After formulating in the Berggren basis the two-particle problem and
the particle-rotor model, a general multi-particle configuration mixing approach
of the Gamow shell model in the Slater determinants representation is discussed
in detail. Its mathematical setting in the rigged Hilbert space, the many-body
Berggren relation, and the interpretational issues related to complex observables are
presented. Gamow shell model can be formulated either in the core-plus-valenceparticles model space using the translationally invariant scheme of cluster orbital
shell model or, as will be discussed in Chap. 8, having all particles active. In this
chapter, most part of the discussion deals with the first option.
The optimization of single-particle basis and an appropriate truncation strategy of
many-body Berggren basis in an extended model space, including the single-particle
shells of discretized continuum, are the topics of great importance for any practical
application of the Gamow shell model. The diagonalization of large Gamow shell
model matrices is discussed using the density matrix renormalization approach and
the Jacobi-Davidson method. Related to that is the issue of calculation of the twoparticle matrix elements and their optimal storage in the large-scale calculations.
A separate important problem is the quantification of theoretical errors on
quantities calculated in the Gamow shell model. The statistical evaluation of the
quality of the Gamow shell model predictions includes calculation of the penalty
function and Bayesian inference of parameters.
Chapter 6 How the matrix elements of the effective interaction in Gamow shell
model depend on angular momentum and continuum coupling is an issue of
great importance for understanding of the complex spectra in atomic nuclei. Do
they follow in weakly bound states or resonances a similar dependence as found
in a standard shell model for bound states? This important topic is addressed
by analyzing the dependence of renormalized two-body matrix elements on the
semiclassical angle between angular momenta of the coupled nucleons, and by
calculating the two-particle spectra in the shells of different angular momenta.
Moreover, selected examples of Gamow shell model spectra in light nuclei are
discussed and compared with the experimental information.
The Borromean halo nuclei and the emitters of two-protons or two-neutrons
are intriguing examples of near-threshold phenomena. One can grasp the structure
of these near-threshold states either in the Gamow three-body model in Jacobi
relative coordinates, or in the Gamow shell model in the cluster orbital shell model
coordinates. Comparison of these approaches on selected observables provides the
convergence test for these two Berggren bases.
Residual pairing interaction and the mean field are essential ingredients of our
understanding of the shell structure and the properties of heavy nuclei. In the vicinity
of continuum, the role of pairing becomes essential. Exact solution of the pairing
Hamiltonian in a schematic model exists for bound single-particle levels [66, 67].
The generalization of this solution in Berggren basis may help to describe pairing
correlations in systems with a large number of fermions, such as the ultra-small
superconducting grains or nuclei with large number of valence particles, which
cannot be discussed in the Gamow shell model applications.
1 Introduction: From Bound States to the Continuum
Chapter 5 After formulating in the Berggren basis the two-particle problem and
the particle-rotor model, a general multi-particle configuration mixing approach
of the Gamow shell model in the Slater determinants representation is discussed
in detail. Its mathematical setting in the rigged Hilbert space, the many-body
Berggren relation, and the interpretational issues related to complex observables are
presented. Gamow shell model can be formulated either in the core-plus-valenceparticles model space using the translationally invariant scheme of cluster orbital
shell model or, as will be discussed in Chap. 8, having all particles active. In this
chapter, most part of the discussion deals with the first option.
The optimization of single-particle basis and an appropriate truncation strategy of
many-body Berggren basis in an extended model space, including the single-particle
shells of discretized continuum, are the topics of great importance for any practical
application of the Gamow shell model. The diagonalization of large Gamow shell
model matrices is discussed using the density matrix renormalization approach and
the Jacobi-Davidson method. Related to that is the issue of calculation of the twoparticle matrix elements and their optimal storage in the large-scale calculations.
A separate important problem is the quantification of theoretical errors on
quantities calculated in the Gamow shell model. The statistical evaluation of the
quality of the Gamow shell model predictions includes calculation of the penalty
function and Bayesian inference of parameters.
Chapter 6 How the matrix elements of the effective interaction in Gamow shell
model depend on angular momentum and continuum coupling is an issue of
great importance for understanding of the complex spectra in atomic nuclei. Do
they follow in weakly bound states or resonances a similar dependence as found
in a standard shell model for bound states? This important topic is addressed
by analyzing the dependence of renormalized two-body matrix elements on the
semiclassical angle between angular momenta of the coupled nucleons, and by
calculating the two-particle spectra in the shells of different angular momenta.
Moreover, selected examples of Gamow shell model spectra in light nuclei are
discussed and compared with the experimental information.
The Borromean halo nuclei and the emitters of two-protons or two-neutrons
are intriguing examples of near-threshold phenomena. One can grasp the structure
of these near-threshold states either in the Gamow three-body model in Jacobi
relative coordinates, or in the Gamow shell model in the cluster orbital shell model
coordinates. Comparison of these approaches on selected observables provides the
convergence test for these two Berggren bases.
Residual pairing interaction and the mean field are essential ingredients of our
understanding of the shell structure and the properties of heavy nuclei. In the vicinity
of continuum, the role of pairing becomes essential. Exact solution of the pairing
Hamiltonian in a schematic model exists for bound single-particle levels [66, 67].
The generalization of this solution in Berggren basis may help to describe pairing
correlations in systems with a large number of fermions, such as the ultra-small
superconducting grains or nuclei with large number of valence particles, which
cannot be discussed in the Gamow shell model applications.