372
8 No-Core Gamow Shell Model
system, one needs 3A − 3 coordinates, whereas the constructed wave function,
depends on 3A coordinates. The redundant degrees of freedom are responsible
for the spurious center-of-mass excitations. Plane wave single-particle states are
eigenstates of the momentum operator and, hence, preserve the translational invariance but, unfortunately, cannot be used to describe a localized system. The standard
method to suppress spurious center-of-mass excitations is to solve the many-body
problem using relative coordinates, e.g. Jacobi coordinates. This ensures that the
translational invariance of the system is preserved. However, antisymmetrization
of states becomes unfeasible for A ≥ 6, typically, due to the current limitations
of computing resources. Consequently, an alternative strategy is considered for
practical applications. For this, one uses the unique analytical properties of the
harmonic oscillator single-particle basis in a full N ¯
hω space, in which the total wave
function factorizes into |ψ rel ⊗ |ψ CM . However, due to the confining character of
the harmonic oscillator basis states, this limits the application of this method to
well-bound systems only.
In the case of the no-core Gamow shell model, the latter factorization is not
guaranteed. Since the Hamiltonian of Eq. (8.1) is expressed in relative coordinates
of nucleons, one is expecting that eigenstates would not contain spurious center-ofmass excitations in an infinite space. However, because one is working in a finite
space, it is necessary to demonstrate numerically this condition. Assuming that the
factorization into a center-of-mass and a relative wave functions is valid and also
the center-of-mass wave function has a Gaussian shape, the expectation value of the
center-of-mass operator is [91]
ˆ
H CM =
1
2mA
P
2
CM +
mAω 2
2
R
2
CM −
3
2
¯
hω ,
(8.6)
where ¯
hω is the parameter that characterizes the Gaussian wave function. The matrix
elements of (8.6) are calculated with the harmonic oscillator expansion method (see
Sect. 5.6 and Ref. [92]).
Following the assertion of Ref. [93], if ˆ
H CM ∼ 0 then the factorization is
valid (see Sect. 8.2.1 for a detailed study of the center-of-mass correction in the
no-core Gamow shell model). However, this condition is useful only for bound
states. Indeed, it has been noticed that the use of the Lawson method for many-body
resonances may bind these states. Moreover, (8.6) is the only two-body center-ofmass Hamiltonian that can be applied in shell model applications with the Lawson
method. The only other available two-body center-of-mass Hamiltonian is P 2
CM ,
which possesses no ground state as all its eigenstates are center-of-mass Bessel
functions. Consequently, there is a priori no method to fix the center-of-mass of
an unbound state.
8 No-Core Gamow Shell Model
system, one needs 3A − 3 coordinates, whereas the constructed wave function,
depends on 3A coordinates. The redundant degrees of freedom are responsible
for the spurious center-of-mass excitations. Plane wave single-particle states are
eigenstates of the momentum operator and, hence, preserve the translational invariance but, unfortunately, cannot be used to describe a localized system. The standard
method to suppress spurious center-of-mass excitations is to solve the many-body
problem using relative coordinates, e.g. Jacobi coordinates. This ensures that the
translational invariance of the system is preserved. However, antisymmetrization
of states becomes unfeasible for A ≥ 6, typically, due to the current limitations
of computing resources. Consequently, an alternative strategy is considered for
practical applications. For this, one uses the unique analytical properties of the
harmonic oscillator single-particle basis in a full N ¯
hω space, in which the total wave
function factorizes into |ψ rel ⊗ |ψ CM . However, due to the confining character of
the harmonic oscillator basis states, this limits the application of this method to
well-bound systems only.
In the case of the no-core Gamow shell model, the latter factorization is not
guaranteed. Since the Hamiltonian of Eq. (8.1) is expressed in relative coordinates
of nucleons, one is expecting that eigenstates would not contain spurious center-ofmass excitations in an infinite space. However, because one is working in a finite
space, it is necessary to demonstrate numerically this condition. Assuming that the
factorization into a center-of-mass and a relative wave functions is valid and also
the center-of-mass wave function has a Gaussian shape, the expectation value of the
center-of-mass operator is [91]
ˆ
H CM =
1
2mA
P
2
CM +
mAω 2
2
R
2
CM −
3
2
¯
hω ,
(8.6)
where ¯
hω is the parameter that characterizes the Gaussian wave function. The matrix
elements of (8.6) are calculated with the harmonic oscillator expansion method (see
Sect. 5.6 and Ref. [92]).
Following the assertion of Ref. [93], if ˆ
H CM ∼ 0 then the factorization is
valid (see Sect. 8.2.1 for a detailed study of the center-of-mass correction in the
no-core Gamow shell model). However, this condition is useful only for bound
states. Indeed, it has been noticed that the use of the Lawson method for many-body
resonances may bind these states. Moreover, (8.6) is the only two-body center-ofmass Hamiltonian that can be applied in shell model applications with the Lawson
method. The only other available two-body center-of-mass Hamiltonian is P 2
CM ,
which possesses no ground state as all its eigenstates are center-of-mass Bessel
functions. Consequently, there is a priori no method to fix the center-of-mass of
an unbound state.