5
Formulation and Implementation of the
Gamow Shell Model
5.1
Mathematical Foundation of the Gamow Shell Model
The Gamow shell model formulated in the Berggren basis [1] has all necessary
features which allow to describe correlated A-body halo states and resonances.
In contrast to the standard shell model, the Gamow shell model respects the flux
conservation, i.e., provides unitary description of wave functions in the vicinity and
above elastic reaction threshold. By using a configuration mixing framework, the
Gamow shell model bears the same generality as the standard shell model, providing
a microscopic description of all kinds of nuclei. However, due to the use of unbound
states in the one-body basis, a rigorous mathematical foundation of the Gamow shell
model is more complicated because resonances do not belong to the Hilbert space.
The Berggren basis has been introduced at the one-body level using the Newton
completeness relation and Cauchy theorem. Completeness of this basis has then
been demonstrated in a most general case using the analytical properties of Coulomb
wave functions and the Cauchy theorem. However, if we intend to use the Berggren
basis in shell model, then it becomes very cumbersome to continue using onebody completeness relations in the coordinate space (r-space), as shell model
wave functions are multidimensional objects. Moreover, a feature as simple as the
antisymmetry, which is straightforward to apply using Slater determinants, becomes
very complicated to handle in coordinate space. Consequently, if one wants to keep
the convenience of the shell model framework, it is necessary to use the abstract
formulation of second quantization, based on the use of creation and annihilation
operators on quantum bra and ket states.
Intuitively, one can already suppose that the algebra needed for the Gamow shell
model will be very close to that used in the Hilbert space. Indeed, if one heuristically
replaces the discrete completeness relation of bound states by a continuous one
involving integration of bra and ket of unbound states, which one discretizes
afterward, one obtains formulas which represent a multidimensional generalized
Fourier transform. However, using Dirac notation with unbound states would remain
© Springer International Publishing AG 2021
N. Michel, M. Płoszajczak, Gamow Shell Model, Lecture Notes in Physics 983,
https://doi.org/10.1007/978-3-030-69356-5_5
185
Formulation and Implementation of the
Gamow Shell Model
5.1
Mathematical Foundation of the Gamow Shell Model
The Gamow shell model formulated in the Berggren basis [1] has all necessary
features which allow to describe correlated A-body halo states and resonances.
In contrast to the standard shell model, the Gamow shell model respects the flux
conservation, i.e., provides unitary description of wave functions in the vicinity and
above elastic reaction threshold. By using a configuration mixing framework, the
Gamow shell model bears the same generality as the standard shell model, providing
a microscopic description of all kinds of nuclei. However, due to the use of unbound
states in the one-body basis, a rigorous mathematical foundation of the Gamow shell
model is more complicated because resonances do not belong to the Hilbert space.
The Berggren basis has been introduced at the one-body level using the Newton
completeness relation and Cauchy theorem. Completeness of this basis has then
been demonstrated in a most general case using the analytical properties of Coulomb
wave functions and the Cauchy theorem. However, if we intend to use the Berggren
basis in shell model, then it becomes very cumbersome to continue using onebody completeness relations in the coordinate space (r-space), as shell model
wave functions are multidimensional objects. Moreover, a feature as simple as the
antisymmetry, which is straightforward to apply using Slater determinants, becomes
very complicated to handle in coordinate space. Consequently, if one wants to keep
the convenience of the shell model framework, it is necessary to use the abstract
formulation of second quantization, based on the use of creation and annihilation
operators on quantum bra and ket states.
Intuitively, one can already suppose that the algebra needed for the Gamow shell
model will be very close to that used in the Hilbert space. Indeed, if one heuristically
replaces the discrete completeness relation of bound states by a continuous one
involving integration of bra and ket of unbound states, which one discretizes
afterward, one obtains formulas which represent a multidimensional generalized
Fourier transform. However, using Dirac notation with unbound states would remain
© Springer International Publishing AG 2021
N. Michel, M. Płoszajczak, Gamow Shell Model, Lecture Notes in Physics 983,
https://doi.org/10.1007/978-3-030-69356-5_5
185