298
6 Physical Applications of the Gamow Shell Model
6.6.2.2 Dependence of Observables on the Rotation Angle in Practical
Calculations
While eigenenergies are theoretically independent of θ due to the ABC theorem,
a spurious θ dependence arises from the use of a finite model space. It is then
necessary in practice to determine the optimal angle θ opt with which truncation
errors are minimal. For this, one uses the generalized variational principle, described
in Sect. 5.1.4. Indeed, eigenenergies calculated with the rotation angle θ opt are
stationary with respect to small changes in θ . The determination of θ opt is straightforward, as it is sufficient for that matter to diagonalize ˆ
H θ with θ min < θ < π/4,
where θ min is the smallest rotation angle rendering the sought resonance state
integrable on the real r-axis.
Another issue inherent to the use of Hamiltonian complex scaling method in
a finite model space is the instability of wave function back-transformation, i.e.
the application of ˆ
U
−1
θ
(see Eq. (6.66)) on rotated wave functions to obtain the
unbound eigenstates of ˆ
H [112, 160, 161]. This problem occurs when one wants to
calculate wave function observables which explicitly depend on r, such as density or
correlation density [112, 160]. The situation is similar to inverse Fourier transform,
which is notoriously known to be unstable numerically [162].
Indeed, the application of ˆ
U
−1
θ
(see Eq. (6.66)) to the rotated resonance wave
function, which is integrable, yields the physical resonance wave function on the real
r-axis which diverges exponentially in modulus. Consequently, the small numerical
inaccuracies present in the rotated resonance wave function are exponentially
amplified in this process. In fact, it has been proved that the back-transformed
resonance wave function calculated in a truncated space, in general does not
converge to the exact resonance wave function at the limit of infinitely large model
space [160]. Moreover, as could be expected, the numerical error increases along
with θ [112].
As for inverse Fourier transform, this problem could be solved by using a
priori knowledge on the eigenstate [112, 162]. Indeed, as one knows that the
exact resonance wave function has smooth variations on the real r-axis, it can be
reconstructed by suppressing spurious high frequency components. This has been
realized in Ref. [112] using the Tikhonov regularization method [163]. It was shown
that the density of the unbound 2 + first excited state of 6 He, modelled by two
valence neutrons above a 4 He core, could be calculated accurately [112].
The Tikhonov regularization method depends on a regularization parameter,
which is determined by way of a plateau condition [112]. For this, one determines
the optimal regularization parameter for which the variations of density with respect
to this parameter are minimal [112]. The precision of the Tikhonov regularization
method could be assessed from Gamow shell model calculations, with which the
density of resonance states can be almost exactly calculated [112]. Note that rintegrated observables, such as the angular correlation density, do not suffer from
this instability and can be accurately calculated with the Hamiltonian complex
scaling [112].
6 Physical Applications of the Gamow Shell Model
6.6.2.2 Dependence of Observables on the Rotation Angle in Practical
Calculations
While eigenenergies are theoretically independent of θ due to the ABC theorem,
a spurious θ dependence arises from the use of a finite model space. It is then
necessary in practice to determine the optimal angle θ opt with which truncation
errors are minimal. For this, one uses the generalized variational principle, described
in Sect. 5.1.4. Indeed, eigenenergies calculated with the rotation angle θ opt are
stationary with respect to small changes in θ . The determination of θ opt is straightforward, as it is sufficient for that matter to diagonalize ˆ
H θ with θ min < θ < π/4,
where θ min is the smallest rotation angle rendering the sought resonance state
integrable on the real r-axis.
Another issue inherent to the use of Hamiltonian complex scaling method in
a finite model space is the instability of wave function back-transformation, i.e.
the application of ˆ
U
−1
θ
(see Eq. (6.66)) on rotated wave functions to obtain the
unbound eigenstates of ˆ
H [112, 160, 161]. This problem occurs when one wants to
calculate wave function observables which explicitly depend on r, such as density or
correlation density [112, 160]. The situation is similar to inverse Fourier transform,
which is notoriously known to be unstable numerically [162].
Indeed, the application of ˆ
U
−1
θ
(see Eq. (6.66)) to the rotated resonance wave
function, which is integrable, yields the physical resonance wave function on the real
r-axis which diverges exponentially in modulus. Consequently, the small numerical
inaccuracies present in the rotated resonance wave function are exponentially
amplified in this process. In fact, it has been proved that the back-transformed
resonance wave function calculated in a truncated space, in general does not
converge to the exact resonance wave function at the limit of infinitely large model
space [160]. Moreover, as could be expected, the numerical error increases along
with θ [112].
As for inverse Fourier transform, this problem could be solved by using a
priori knowledge on the eigenstate [112, 162]. Indeed, as one knows that the
exact resonance wave function has smooth variations on the real r-axis, it can be
reconstructed by suppressing spurious high frequency components. This has been
realized in Ref. [112] using the Tikhonov regularization method [163]. It was shown
that the density of the unbound 2 + first excited state of 6 He, modelled by two
valence neutrons above a 4 He core, could be calculated accurately [112].
The Tikhonov regularization method depends on a regularization parameter,
which is determined by way of a plateau condition [112]. For this, one determines
the optimal regularization parameter for which the variations of density with respect
to this parameter are minimal [112]. The precision of the Tikhonov regularization
method could be assessed from Gamow shell model calculations, with which the
density of resonance states can be almost exactly calculated [112]. Note that rintegrated observables, such as the angular correlation density, do not suffer from
this instability and can be accurately calculated with the Hamiltonian complex
scaling [112].