3.7 Numerical Implementation of the Berggren Completeness Relation
123
This curious feature of the off-diagonal method can be explained by noticing
that the integrand of the first integral on the right-hand side of Eq. (3.93), which is
formed by radial integrals converging with the complex rotation of r (see Eq. (3.90)),
while everywhere finite, is not analytic at k ∼ k. Indeed, it is equivalent, up
to an unimportant constant, to (k − k ) ln(k − k ), which does not even possess
a finite derivative with respect to k at k = k. Consequently, a Gauss-Legendre
discretization of this integral will be far less precise than that of analytic functions,
which can be usually well approximated by polynomials. On the contrary, the use
of u
+
i (r) and u
−
i (r) to calculate the diagonal matrix element of V Coul (ΔZ, r),
effectively replaces the matrix elements involving u(k, r) and u(k , r) with k ∼ k
by an analytic function of k. Indeed, the moduli of matrix elements involving u(k, r)
and u(k , r) with k ∼ k are very large, because matrix elements have a singularity
in k = k due to the infinite range of the Coulomb potential. Conversely, the analytic
function of k replacing these matrix elements has smooth variations when k ∼ k.
Therefore, the Gauss-Legendre quadrature yields fast convergence to the numerical
value of the integral of Eq. (3.95). The efficiency of the off-diagonal method in
practical applications is considered in more details in Exercise XIII.
Exercise XIII
Based on numerical examples, one will show that the off-diagonal method is
efficient for practical applications.
Recalculate previously shown results with the one-particle diagonalization
code.
Change the number of protons of the Coulomb potential of the diagonalized
Woods-Saxon potential by a few units. Notice that the Berggren basis expansion
of radial wave functions is precise in this case.
Explain why the numerical precision of the Berggren basis expansion of radial
wave functions in the presence of a Coulomb potential is independent of the charge
defining the Coulomb potential.
3.7.2 Completeness Relations Involving Antibound s 1/2 States
An important question in the Gamow shell model concerns the inclusion of
antibound states in the Berggren ensemble [49]. Antibound states have real and
negative energy eigenvalues that are located in the second Riemann sheet of the
complex energy plane (the corresponding momentum lies on the negative imaginary
k-axis) [9, 50–52]. Contrary to bound states, the radial wave functions of antibound
states increase exponentially at large distances. As often discussed in the literature,
it is difficult to give a clear physical interpretation to antibound states. Strictly
speaking, as the second energy sheet is unphysical and inaccessible for direct
experiments, the antibound pole of the scattering matrix is not actually a state but
rather a feature of the system. Antibound states with small energy greatly increase
a localization of real-energy scattering states just above the threshold [53, 54].
123
This curious feature of the off-diagonal method can be explained by noticing
that the integrand of the first integral on the right-hand side of Eq. (3.93), which is
formed by radial integrals converging with the complex rotation of r (see Eq. (3.90)),
while everywhere finite, is not analytic at k ∼ k. Indeed, it is equivalent, up
to an unimportant constant, to (k − k ) ln(k − k ), which does not even possess
a finite derivative with respect to k at k = k. Consequently, a Gauss-Legendre
discretization of this integral will be far less precise than that of analytic functions,
which can be usually well approximated by polynomials. On the contrary, the use
of u
+
i (r) and u
−
i (r) to calculate the diagonal matrix element of V Coul (ΔZ, r),
effectively replaces the matrix elements involving u(k, r) and u(k , r) with k ∼ k
by an analytic function of k. Indeed, the moduli of matrix elements involving u(k, r)
and u(k , r) with k ∼ k are very large, because matrix elements have a singularity
in k = k due to the infinite range of the Coulomb potential. Conversely, the analytic
function of k replacing these matrix elements has smooth variations when k ∼ k.
Therefore, the Gauss-Legendre quadrature yields fast convergence to the numerical
value of the integral of Eq. (3.95). The efficiency of the off-diagonal method in
practical applications is considered in more details in Exercise XIII.
Exercise XIII
Based on numerical examples, one will show that the off-diagonal method is
efficient for practical applications.
Recalculate previously shown results with the one-particle diagonalization
code.
Change the number of protons of the Coulomb potential of the diagonalized
Woods-Saxon potential by a few units. Notice that the Berggren basis expansion
of radial wave functions is precise in this case.
Explain why the numerical precision of the Berggren basis expansion of radial
wave functions in the presence of a Coulomb potential is independent of the charge
defining the Coulomb potential.
3.7.2 Completeness Relations Involving Antibound s 1/2 States
An important question in the Gamow shell model concerns the inclusion of
antibound states in the Berggren ensemble [49]. Antibound states have real and
negative energy eigenvalues that are located in the second Riemann sheet of the
complex energy plane (the corresponding momentum lies on the negative imaginary
k-axis) [9, 50–52]. Contrary to bound states, the radial wave functions of antibound
states increase exponentially at large distances. As often discussed in the literature,
it is difficult to give a clear physical interpretation to antibound states. Strictly
speaking, as the second energy sheet is unphysical and inaccessible for direct
experiments, the antibound pole of the scattering matrix is not actually a state but
rather a feature of the system. Antibound states with small energy greatly increase
a localization of real-energy scattering states just above the threshold [53, 54].