6.6 Comparison of Gamow Shell Model and Hamiltonian Complex Scaling. . .
297
operator of Eq. (6.66) makes resonance states integrable, the exponential decay rate
of complex-rotated wave functions can be small. For example, in the one-body case,
it is equal to |k| at most, so that weakly unbound states decay slowly on the real raxis. Consequently, convergence would also be very slow if one uses a basis of
harmonic oscillator states to diagonalize ˆ
H θ in order to evaluate weakly unbound
resonance states (see Eq. (6.67)).
As a matter of fact, a basis of harmonic oscillator states, and in general a basis of
bound states generated by a potential which goes to infinity with increasing r, is not
suitable to calculate weakly bound and resonance many-body states. In principle,
one could recur to the Newton completeness relation generated by a finite-range
potential, as it has the same asymptotic properties as Fourier transform, so that
slowly decreasing functions can be expanded (see Sect. 3.2.3). However, this would
amount to utilize the Gamow shell model formalism, as one would have to discretize
the continuum of the Newton completeness relation similarly to that of the Berggren
basis (see Sect. 3.7). The whole point of using a complex-scaled Hamiltonian would
then be forfeited, as its fundamental interest is to allow the use of discrete basis of
integrable states for Hamiltonian diagonalization.
It is possible to solve this conundrum by using a non-orthogonal set of basis
states. A widely used basis of this type is the Gaussian basis, where one uses
Gaussian form factors of different lengths [147–149,152]. The numerical advantage
of Gaussian bases is twofold. On the one hand, convergence for weakly bound and
resonance states is obtained with a fairly small number of basis states, because the
Gaussian functions of the used basis have different spatial extensions. Moreover,
the simple mathematical form of Gaussian basis states often allows to evaluate
analytically two-body matrix elements, for example, if the used interaction is of
Coulomb or Gaussian type. For nuclear applications, it is especially useful when
the Hamiltonian consists of a KKNN potential mimicking a core and of a residual
Gaussian force, as then Hamiltonian matrix elements can be calculated analytically.
The many-body Schrödinger equation becomes a generalized eigenvalue problem when representing the Hamiltonian with a non-orthogonal set of basis states,
and is of the form:
ˆ
H θ |Ψ = E ˆ
O |Ψ ,
where ˆ
O is the basis overlap matrix. The generalized eigenvalue problem is
straightforward to solve when dimensions reach few thousands at most, as it can
be reduced to the standard eigenvalue problem
ˆ
O
−1/2 ˆ
H θ ˆ
O
−1/2
|Φ = E |Φ ,
with |Ψ = ˆ
O −1/2 |Φ. As the ˆ
O −1/2 operator is straightforward to calculate
from the eigenvalues and eigenvectors of ˆ
O, the numerical cost to solve a generalized eigenvalue problem is only a few times slower than that of a standard
eigenvalue problem. Otherwise, it is possible to use extensions of the Lanczos or
Jacobi–Davidson methods to the generalized eigenvalue problem, which have been
developed to deal with matrices of large dimensions [157–159].
297
operator of Eq. (6.66) makes resonance states integrable, the exponential decay rate
of complex-rotated wave functions can be small. For example, in the one-body case,
it is equal to |k| at most, so that weakly unbound states decay slowly on the real raxis. Consequently, convergence would also be very slow if one uses a basis of
harmonic oscillator states to diagonalize ˆ
H θ in order to evaluate weakly unbound
resonance states (see Eq. (6.67)).
As a matter of fact, a basis of harmonic oscillator states, and in general a basis of
bound states generated by a potential which goes to infinity with increasing r, is not
suitable to calculate weakly bound and resonance many-body states. In principle,
one could recur to the Newton completeness relation generated by a finite-range
potential, as it has the same asymptotic properties as Fourier transform, so that
slowly decreasing functions can be expanded (see Sect. 3.2.3). However, this would
amount to utilize the Gamow shell model formalism, as one would have to discretize
the continuum of the Newton completeness relation similarly to that of the Berggren
basis (see Sect. 3.7). The whole point of using a complex-scaled Hamiltonian would
then be forfeited, as its fundamental interest is to allow the use of discrete basis of
integrable states for Hamiltonian diagonalization.
It is possible to solve this conundrum by using a non-orthogonal set of basis
states. A widely used basis of this type is the Gaussian basis, where one uses
Gaussian form factors of different lengths [147–149,152]. The numerical advantage
of Gaussian bases is twofold. On the one hand, convergence for weakly bound and
resonance states is obtained with a fairly small number of basis states, because the
Gaussian functions of the used basis have different spatial extensions. Moreover,
the simple mathematical form of Gaussian basis states often allows to evaluate
analytically two-body matrix elements, for example, if the used interaction is of
Coulomb or Gaussian type. For nuclear applications, it is especially useful when
the Hamiltonian consists of a KKNN potential mimicking a core and of a residual
Gaussian force, as then Hamiltonian matrix elements can be calculated analytically.
The many-body Schrödinger equation becomes a generalized eigenvalue problem when representing the Hamiltonian with a non-orthogonal set of basis states,
and is of the form:
ˆ
H θ |Ψ = E ˆ
O |Ψ ,
where ˆ
O is the basis overlap matrix. The generalized eigenvalue problem is
straightforward to solve when dimensions reach few thousands at most, as it can
be reduced to the standard eigenvalue problem
ˆ
O
−1/2 ˆ
H θ ˆ
O
−1/2
|Φ = E |Φ ,
with |Ψ = ˆ
O −1/2 |Φ. As the ˆ
O −1/2 operator is straightforward to calculate
from the eigenvalues and eigenvectors of ˆ
O, the numerical cost to solve a generalized eigenvalue problem is only a few times slower than that of a standard
eigenvalue problem. Otherwise, it is possible to use extensions of the Lanczos or
Jacobi–Davidson methods to the generalized eigenvalue problem, which have been
developed to deal with matrices of large dimensions [157–159].