5.1 Mathematical Foundation of the Gamow Shell Model
193
down in the complex plane by a factor larger than 10 compared to the initial
peak.
Notice that the 6 He eigenstates are poorly described with Berggren basis
contours either too close or too far from the 0p 3/2 basis resonance state for
a fixed contour discretization, whereas the initial contour provided with very
accurate results.
Show that increasing the number of discretized scattering states does not
ameliorate the situation when the contour is too close to the 0p 3/2 resonance
state.
Alternatively, show that the number of discretized scattering states needed to
obtain an acceptable precision becomes prohibitively large when the contour
is too far from the 0p 3/2 resonance state.
C. Discuss the imprecision induced by the discretization of Berggren scattering
contours and phase shift variation along the p 3/2 contour.
Explain why optimal contours exist for the numerical implementation of the
Berggren completeness relation, with which the number of discretized points
needed to have precise results is minimized.
5.1.4 Complex Observables: The Generalized Variational Principle
The Gamow shell model Hamiltonian is represented by a complex symmetric
matrix, which possess in general complex eigenvalues. The most important of them
are those corresponding to resonance states, as the real part of the eigenvalue
is interpreted as its energy and its imaginary part as minus half its width (see
Sect. 2.6.6). However, one cannot assess for the moment the validity of the
calculation of a many-body resonance state in a finite model space. Indeed, as its
eigenvalue is complex, the variational principle cannot apply. Another method to
state how precise the calculated eigenvector is compared to the exact eigenstate of
the Hamiltonian is thus needed. For this, one relies on the generalized variational
principle.
Indeed, the analytic continuation of the equations of the variational principle
leads to the Gamow shell model eigenproblem in a finite model space. As one deals
with complex energies, one will firstly vary their squared modulus in order to find
the stationary point a real-valued function:
δ(|E(Ψ )|
2 ) = 0
⇔ E
∗ (Ψ )δE(Ψ ) + E(Ψ )δE
∗ (Ψ ) = 0 .
(5.21)
what implies
δE(Ψ ) = 0
193
down in the complex plane by a factor larger than 10 compared to the initial
peak.
Notice that the 6 He eigenstates are poorly described with Berggren basis
contours either too close or too far from the 0p 3/2 basis resonance state for
a fixed contour discretization, whereas the initial contour provided with very
accurate results.
Show that increasing the number of discretized scattering states does not
ameliorate the situation when the contour is too close to the 0p 3/2 resonance
state.
Alternatively, show that the number of discretized scattering states needed to
obtain an acceptable precision becomes prohibitively large when the contour
is too far from the 0p 3/2 resonance state.
C. Discuss the imprecision induced by the discretization of Berggren scattering
contours and phase shift variation along the p 3/2 contour.
Explain why optimal contours exist for the numerical implementation of the
Berggren completeness relation, with which the number of discretized points
needed to have precise results is minimized.
5.1.4 Complex Observables: The Generalized Variational Principle
The Gamow shell model Hamiltonian is represented by a complex symmetric
matrix, which possess in general complex eigenvalues. The most important of them
are those corresponding to resonance states, as the real part of the eigenvalue
is interpreted as its energy and its imaginary part as minus half its width (see
Sect. 2.6.6). However, one cannot assess for the moment the validity of the
calculation of a many-body resonance state in a finite model space. Indeed, as its
eigenvalue is complex, the variational principle cannot apply. Another method to
state how precise the calculated eigenvector is compared to the exact eigenstate of
the Hamiltonian is thus needed. For this, one relies on the generalized variational
principle.
Indeed, the analytic continuation of the equations of the variational principle
leads to the Gamow shell model eigenproblem in a finite model space. As one deals
with complex energies, one will firstly vary their squared modulus in order to find
the stationary point a real-valued function:
δ(|E(Ψ )|
2 ) = 0
⇔ E
∗ (Ψ )δE(Ψ ) + E(Ψ )δE
∗ (Ψ ) = 0 .
(5.21)
what implies
δE(Ψ ) = 0