110
3 Berggren Basis and Completeness Relations
be generated by deforming the real k-axis in many different ways, which will result
in different discrete parts as that of Eq. (3.79) (see Ref. [34] for the derivation of
overcompleteness relations similar to Eq. (3.79). In particular, Eq. (3.79) reduces to
Eq. (3.78) when expanded functions vanish after a radius R c [34].
One can see that the number of antibound states in Eq. (3.79) can be reduced
by using a sufficiently small K value compared to that of Eq. (3.78). However,
the continuum part of Eq. (3.79) does not vanish in general for not truncated radial
functions, even for large K values [34]. Thus, similarly to Eqs. (3.78) and (3.79) is
mainly interesting from a theoretical point of view. Indeed, even though expanded
radial function can be extended along the real r-axis, the overcomplete character
of Eq. (3.79) and the fact that its continuum part is sizable renders its applicability
very difficult. In fact, the orthogonal character of the basis states as well as the
possibility to include only narrow or mildly broad resonance states in a discrete
sum of Eq. (3.66) made the Berggren basis both physically relevant and numerically
stable in Gamow shell model applications.
3.6
Newton and Berggren Completeness Relations Generated
by Nonlocal Potentials
Local potentials are the most commonly used potentials in nuclear physics, with, in
particular, the widely used Woods-Saxon potential [35]. Potentials used in atomic
and molecular physics are typically also local, and are deemed as pseudo-potentials
(see Sect. 4.3.1.1 for applications to dipolar and quadrupolar anions). However,
nonlocal potentials often arise in quantum physics as well. This is typically due
to the Pauli principle, whose effects are nonlocal. Hence, the Hartree-Fock potential
generated by a finite range interaction, of nuclear or Coulomb type, is nonlocal
[36]. The Pauli principle also plays an important role in nuclear collisions, so that
nonlocal potentials are frequently used in the reaction theory [37].
However, as one will see in this section, the nonlocal potentials greatly augment
the mathematical difficulties related to the use of one-body states. A nonlocal
Schrödinger equation suitable for physical applications is obtained from Eq. (2.2)
by adding a nonlocal potential to its local part:
v l (r) u(k, r) → v l (r) u(k, r) +
R
r 0
v nl (r, r
) u(k, r
) dr
,
(3.80)
where v nl (r, r ) is a smooth function symmetric in r and r , behaving as r 0 +1 when
r → 0 (same for r ).
Bound states of the nonlocal potential are real, which can be shown using a
method similar to that of Exercise VI in Sect. 2.5. As the non-locality of the HartreeFock potential is induced by a finite range interaction, which plays no role far
from the nucleus or molecule, it is physically justified to have a vanishing nonlocal
potential for r, r ≥ R. One can then show that the boundary conditions verified
3 Berggren Basis and Completeness Relations
be generated by deforming the real k-axis in many different ways, which will result
in different discrete parts as that of Eq. (3.79) (see Ref. [34] for the derivation of
overcompleteness relations similar to Eq. (3.79). In particular, Eq. (3.79) reduces to
Eq. (3.78) when expanded functions vanish after a radius R c [34].
One can see that the number of antibound states in Eq. (3.79) can be reduced
by using a sufficiently small K value compared to that of Eq. (3.78). However,
the continuum part of Eq. (3.79) does not vanish in general for not truncated radial
functions, even for large K values [34]. Thus, similarly to Eqs. (3.78) and (3.79) is
mainly interesting from a theoretical point of view. Indeed, even though expanded
radial function can be extended along the real r-axis, the overcomplete character
of Eq. (3.79) and the fact that its continuum part is sizable renders its applicability
very difficult. In fact, the orthogonal character of the basis states as well as the
possibility to include only narrow or mildly broad resonance states in a discrete
sum of Eq. (3.66) made the Berggren basis both physically relevant and numerically
stable in Gamow shell model applications.
3.6
Newton and Berggren Completeness Relations Generated
by Nonlocal Potentials
Local potentials are the most commonly used potentials in nuclear physics, with, in
particular, the widely used Woods-Saxon potential [35]. Potentials used in atomic
and molecular physics are typically also local, and are deemed as pseudo-potentials
(see Sect. 4.3.1.1 for applications to dipolar and quadrupolar anions). However,
nonlocal potentials often arise in quantum physics as well. This is typically due
to the Pauli principle, whose effects are nonlocal. Hence, the Hartree-Fock potential
generated by a finite range interaction, of nuclear or Coulomb type, is nonlocal
[36]. The Pauli principle also plays an important role in nuclear collisions, so that
nonlocal potentials are frequently used in the reaction theory [37].
However, as one will see in this section, the nonlocal potentials greatly augment
the mathematical difficulties related to the use of one-body states. A nonlocal
Schrödinger equation suitable for physical applications is obtained from Eq. (2.2)
by adding a nonlocal potential to its local part:
v l (r) u(k, r) → v l (r) u(k, r) +
R
r 0
v nl (r, r
) u(k, r
) dr
,
(3.80)
where v nl (r, r ) is a smooth function symmetric in r and r , behaving as r 0 +1 when
r → 0 (same for r ).
Bound states of the nonlocal potential are real, which can be shown using a
method similar to that of Exercise VI in Sect. 2.5. As the non-locality of the HartreeFock potential is induced by a finite range interaction, which plays no role far
from the nucleus or molecule, it is physically justified to have a vanishing nonlocal
potential for r, r ≥ R. One can then show that the boundary conditions verified