214
5 Formulation and Implementation of the Gamow Shell Model
where Greek letters label harmonic oscillator states and N is the number of
harmonic oscillator states used in a given partial wave. This method allows to
calculate the two-body part of the Coulomb interaction with the Talmi-BrodyMoshinsky transformation by separating its finite-range and infinite-range parts.
5.7
The Jacobi-Davidson Method in the Gamow Shell Model
The Lanczos method is the tool of choice in standard shell model as it makes
use of matrix ˆ
H times vector operations only, with which the lowest eigenvalues
converge first. While the Lanczos method could be used for the search of bound
states in Gamow shell model, the Lanczos method does not converge in general
for resonance states. This arises from the presence of numerous scattering states
surrounding any resonance state, with the same quantum numbers as the resonance.
Consequently, it is necessary to use a diagonalization method which is targeting
directly the resonant states. The Jacobi-Davidson method is the best method for
that matter. Indeed, it only involves ˆ
H times vector operations, as in the Lanczos
method, while including an approximated shift-and-invert method, so that it quickly
provides with eigenvalues of unbound states.
The algorithm of the Jacobi-Davidson method implemented in the Gamow shell
model is shortly described in the following (see Ref. [16] for an introduction to this
method):
• Start from an approximation of the Gamow shell model eigenvector |Ψ i
The first eigenvector |Ψ 0 comes from pole approximation, where the Lanczos
method can be used, or from another calculation.
• Calculate its residue |R i = ˆ
H |Ψ i − E i |Ψ i
• Compute the new Gamow shell model Davidson vector by solving the linear
system: ( ˆ
H app − E i ) |Φ i+1 = |R i , where ˆ
H app is a preconditioner of ˆ
H (see
explanations below).
• |Φ i+1 is orthogonalized with respect to all previous Jacobi-Davidson vectors
|Φ j , 0 ≤ j ≤ i, then projected on J if necessary (see Sect. 5.8.4), and
normalized.
• ˆ
H is then diagonalized in the set of |Φ j , 0 ≤ j ≤ i + 1, which provides with a
new approximation of the Gamow shell model eigenvalue and eigenvector E i+1
and |Ψ i+1
• Iterate until convergence is reached.
The fundamental problem of the Jacobi-Davidson method is to find a good
preconditioner ˆ
H app for ˆ
H . A diagonal approximation for ˆ
H app is not sufficient
therein, as off-diagonal matrix elements are large in the pole space. However, the
diagonal of the matrix of ˆ
H is dominant in the basis space spanned by scattering
Slater determinants. Consequently, it is efficient to take ˆ
H app equal to ˆ
H in the
subspace generated by Slater determinants built from bound and resonance one-
5 Formulation and Implementation of the Gamow Shell Model
where Greek letters label harmonic oscillator states and N is the number of
harmonic oscillator states used in a given partial wave. This method allows to
calculate the two-body part of the Coulomb interaction with the Talmi-BrodyMoshinsky transformation by separating its finite-range and infinite-range parts.
5.7
The Jacobi-Davidson Method in the Gamow Shell Model
The Lanczos method is the tool of choice in standard shell model as it makes
use of matrix ˆ
H times vector operations only, with which the lowest eigenvalues
converge first. While the Lanczos method could be used for the search of bound
states in Gamow shell model, the Lanczos method does not converge in general
for resonance states. This arises from the presence of numerous scattering states
surrounding any resonance state, with the same quantum numbers as the resonance.
Consequently, it is necessary to use a diagonalization method which is targeting
directly the resonant states. The Jacobi-Davidson method is the best method for
that matter. Indeed, it only involves ˆ
H times vector operations, as in the Lanczos
method, while including an approximated shift-and-invert method, so that it quickly
provides with eigenvalues of unbound states.
The algorithm of the Jacobi-Davidson method implemented in the Gamow shell
model is shortly described in the following (see Ref. [16] for an introduction to this
method):
• Start from an approximation of the Gamow shell model eigenvector |Ψ i
The first eigenvector |Ψ 0 comes from pole approximation, where the Lanczos
method can be used, or from another calculation.
• Calculate its residue |R i = ˆ
H |Ψ i − E i |Ψ i
• Compute the new Gamow shell model Davidson vector by solving the linear
system: ( ˆ
H app − E i ) |Φ i+1 = |R i , where ˆ
H app is a preconditioner of ˆ
H (see
explanations below).
• |Φ i+1 is orthogonalized with respect to all previous Jacobi-Davidson vectors
|Φ j , 0 ≤ j ≤ i, then projected on J if necessary (see Sect. 5.8.4), and
normalized.
• ˆ
H is then diagonalized in the set of |Φ j , 0 ≤ j ≤ i + 1, which provides with a
new approximation of the Gamow shell model eigenvalue and eigenvector E i+1
and |Ψ i+1
• Iterate until convergence is reached.
The fundamental problem of the Jacobi-Davidson method is to find a good
preconditioner ˆ
H app for ˆ
H . A diagonal approximation for ˆ
H app is not sufficient
therein, as off-diagonal matrix elements are large in the pole space. However, the
diagonal of the matrix of ˆ
H is dominant in the basis space spanned by scattering
Slater determinants. Consequently, it is efficient to take ˆ
H app equal to ˆ
H in the
subspace generated by Slater determinants built from bound and resonance one-