250
6 Physical Applications of the Gamow Shell Model
While the generic dependence of (V 12 ) with respect to θ is the same as
found in [22], it is remarkable that this is also true for (V 12 ). This intriguing
result can be understood if one recurs to the procedure of complex rotation, as is
explained afterwards. Indeed, the complex nuclear matrix elements are analytic
continuation of the real nuclear matrix elements involving bound single-particle
states (see Sect. 3.3). Consequently, their analytic properties are analogous to that
of one-body resonant states (see Sect. 3.5), whose width increases when energy
increases. As widths are renormalized in the same manner as energies, it is better to
consider (V 12 ) as a function of the strength of the nuclear interaction. Indeed, the
renormalization in Eqs. (6.11–6.13) removes the one-body widths, so that (V 12 )
is identically zero without any nuclear interaction and becomes nonzero when the
nuclear interaction is present.
It is important to state that (V 12 ) does not have to be positive, as is the case
for particle-emission width. Indeed, (V 12 ) is the difference between the width
of a correlated two-nucleon system and the sum of one-body widths, the latter
sum being the two-body system width in an independent-particle picture (see
Eqs. (6.11), (6.12), and (6.13)). In particular, one will typically have (V 12 ) < 0
if the nucleon interaction is attractive. Indeed, the gain in energy arising from
correlations will make the width of the two-nucleon system smaller than its value
in the independent-particle case. Consequently, when the strength of the interaction
augments, (V 12 ) becomes more and more negative. This explains the qualitative
similarity between the curves (V 12 )(θ ) and (V 12 )(θ ), even though the dispersion
of calculated values for different shells and the two-body angular momentum J is
larger in (V 12 )(θ ) than in (V 12 )(θ ). Due to the renormalization of nuclear matrix
elements (see Eqs. (6.8) and (6.11)), real (V 12 ) and imaginary (V 12 ) parts of
two matrix elements are of the same order of magnitude and vary in the interval
[−4:0], independently of the very different initial values of these matrix elements
in each studied two-nucleon system. Consequently, one can see that the two-body
nuclear matrix elements involving resonance states possess similar properties to
those pertaining to bound states, and that both the real and imaginary parts of
two-body nuclear matrix elements behave similarly due to the analytic character
of two-body wave functions in the complex-energy plane.
6.2.2 Influence of Continuum Coupling on Effective Nuclear Matrix
Elements
The previous studies have been done in the context of renormalized nuclear matrix
elements. One divided all initial nuclear matrix elements by their average value in
order to exhibit their semi-classical behavior as a function of θ (see Figs. 6.3 and
6.4). This allowed to point out the common points between matrix elements for
bound and resonance many-body states, as the overall behavior of renormalized
nuclear matrix elements is similar independently of binding energy of the manybody states. However, the averaging procedure in Eqs. (6.10) and (6.13) masks
6 Physical Applications of the Gamow Shell Model
While the generic dependence of (V 12 ) with respect to θ is the same as
found in [22], it is remarkable that this is also true for (V 12 ). This intriguing
result can be understood if one recurs to the procedure of complex rotation, as is
explained afterwards. Indeed, the complex nuclear matrix elements are analytic
continuation of the real nuclear matrix elements involving bound single-particle
states (see Sect. 3.3). Consequently, their analytic properties are analogous to that
of one-body resonant states (see Sect. 3.5), whose width increases when energy
increases. As widths are renormalized in the same manner as energies, it is better to
consider (V 12 ) as a function of the strength of the nuclear interaction. Indeed, the
renormalization in Eqs. (6.11–6.13) removes the one-body widths, so that (V 12 )
is identically zero without any nuclear interaction and becomes nonzero when the
nuclear interaction is present.
It is important to state that (V 12 ) does not have to be positive, as is the case
for particle-emission width. Indeed, (V 12 ) is the difference between the width
of a correlated two-nucleon system and the sum of one-body widths, the latter
sum being the two-body system width in an independent-particle picture (see
Eqs. (6.11), (6.12), and (6.13)). In particular, one will typically have (V 12 ) < 0
if the nucleon interaction is attractive. Indeed, the gain in energy arising from
correlations will make the width of the two-nucleon system smaller than its value
in the independent-particle case. Consequently, when the strength of the interaction
augments, (V 12 ) becomes more and more negative. This explains the qualitative
similarity between the curves (V 12 )(θ ) and (V 12 )(θ ), even though the dispersion
of calculated values for different shells and the two-body angular momentum J is
larger in (V 12 )(θ ) than in (V 12 )(θ ). Due to the renormalization of nuclear matrix
elements (see Eqs. (6.8) and (6.11)), real (V 12 ) and imaginary (V 12 ) parts of
two matrix elements are of the same order of magnitude and vary in the interval
[−4:0], independently of the very different initial values of these matrix elements
in each studied two-nucleon system. Consequently, one can see that the two-body
nuclear matrix elements involving resonance states possess similar properties to
those pertaining to bound states, and that both the real and imaginary parts of
two-body nuclear matrix elements behave similarly due to the analytic character
of two-body wave functions in the complex-energy plane.
6.2.2 Influence of Continuum Coupling on Effective Nuclear Matrix
Elements
The previous studies have been done in the context of renormalized nuclear matrix
elements. One divided all initial nuclear matrix elements by their average value in
order to exhibit their semi-classical behavior as a function of θ (see Figs. 6.3 and
6.4). This allowed to point out the common points between matrix elements for
bound and resonance many-body states, as the overall behavior of renormalized
nuclear matrix elements is similar independently of binding energy of the manybody states. However, the averaging procedure in Eqs. (6.10) and (6.13) masks