82
3 Berggren Basis and Completeness Relations
is preferable for our purpose to directly demonstrate the one-body completeness
relation from their analytical properties. One can then show explicitly that the onebody completeness relation possesses the same properties as the Fourier transform.
Conversely, demonstrating the one-body completeness relation along these lines can
be performed only when potentials have a Coulomb asymptote, as the asymptotic
behavior of scattering states must follow Eq. (2.7). The latter condition is sufficient
for applications as only Coulomb and centrifugal potentials remain in the asymptotic
region, the nuclear part of the Hamiltonian being of short range.
This one-body completeness relation involves real-energy states only. However,
the Gamow shell model is built from the completeness relation of complex energy
eigenstates (2.3), the Berggren completeness relation [7], where resonance and
scattering states enter. The Berggren completeness relation will be demonstrated
in this chapter from the Newton completeness relation of real-energy one-body
states. Generalizations involving the use of complex-valued potentials or nonlocal
potentials will be provided in this chapter as well.
The Berggren completeness relation is a practical tool, aimed to be used
in numerical calculations. Thus, after demonstrating the Berggren completeness
relation, one will demonstrate that it can be efficiently implemented to integrate the
many-body Schrödinger equation and, furthermore discuss numerical examples in
details. Hamiltonians represented in the Berggren basis take the form of complexsymmetric matrices. The numerical methods needed for their diagonalization are
different from the case of Hermitian matrices and are not standard. One will show,
however, that it is possible to diagonalize numerically the complex-symmetric
Hamiltonian matrices almost as efficiently as the real-symmetric or Hermitian
matrices.
Let us emphasize that the following demonstrations of completeness are technical and might be too cumbersome for the reader. In this latter case, we advise the
reader to go directly to Sect. 3.3, where the Berggren completeness relation will be
devised using the previously demonstrated Newton completeness relation.
3.1
Normalization of Gamow Functions
The standard method to normalize scattering wave functions is to use Dirac delta
normalization [8]. In the case of neutral particle, Dirac delta normalization is
immediate to perform as scattering states behave asymptotically as exp(±ikr)
for r → +∞. On the contrary, the appearance of a logarithm in the asymptote
exp(±i(kr − η ln(2kr))) of the charged particle wave function (see Eq. (2.35))
demands care.
One can show from Eq. (2.130) that bound states and scattering states are
orthogonal to each other. However, one has not proved yet that scattering states
are orthonormal. As they are not integrable (see Eq. (2.7)), the orthonormalization
of scattering states has to be understood in a weak sense, that is, with a Dirac delta
3 Berggren Basis and Completeness Relations
is preferable for our purpose to directly demonstrate the one-body completeness
relation from their analytical properties. One can then show explicitly that the onebody completeness relation possesses the same properties as the Fourier transform.
Conversely, demonstrating the one-body completeness relation along these lines can
be performed only when potentials have a Coulomb asymptote, as the asymptotic
behavior of scattering states must follow Eq. (2.7). The latter condition is sufficient
for applications as only Coulomb and centrifugal potentials remain in the asymptotic
region, the nuclear part of the Hamiltonian being of short range.
This one-body completeness relation involves real-energy states only. However,
the Gamow shell model is built from the completeness relation of complex energy
eigenstates (2.3), the Berggren completeness relation [7], where resonance and
scattering states enter. The Berggren completeness relation will be demonstrated
in this chapter from the Newton completeness relation of real-energy one-body
states. Generalizations involving the use of complex-valued potentials or nonlocal
potentials will be provided in this chapter as well.
The Berggren completeness relation is a practical tool, aimed to be used
in numerical calculations. Thus, after demonstrating the Berggren completeness
relation, one will demonstrate that it can be efficiently implemented to integrate the
many-body Schrödinger equation and, furthermore discuss numerical examples in
details. Hamiltonians represented in the Berggren basis take the form of complexsymmetric matrices. The numerical methods needed for their diagonalization are
different from the case of Hermitian matrices and are not standard. One will show,
however, that it is possible to diagonalize numerically the complex-symmetric
Hamiltonian matrices almost as efficiently as the real-symmetric or Hermitian
matrices.
Let us emphasize that the following demonstrations of completeness are technical and might be too cumbersome for the reader. In this latter case, we advise the
reader to go directly to Sect. 3.3, where the Berggren completeness relation will be
devised using the previously demonstrated Newton completeness relation.
3.1
Normalization of Gamow Functions
The standard method to normalize scattering wave functions is to use Dirac delta
normalization [8]. In the case of neutral particle, Dirac delta normalization is
immediate to perform as scattering states behave asymptotically as exp(±ikr)
for r → +∞. On the contrary, the appearance of a logarithm in the asymptote
exp(±i(kr − η ln(2kr))) of the charged particle wave function (see Eq. (2.35))
demands care.
One can show from Eq. (2.130) that bound states and scattering states are
orthogonal to each other. However, one has not proved yet that scattering states
are orthonormal. As they are not integrable (see Eq. (2.7)), the orthonormalization
of scattering states has to be understood in a weak sense, that is, with a Dirac delta