240
6 Physical Applications of the Gamow Shell Model
by the monopole components of the two-body shell model Hamiltonian [1, 3]. To
account for this effect in the standard shell model, one would need to introduce
a neutron-number dependence of the T=0 monopole terms [1] which appears if
one includes the three-body interactions in the framework of a standard shell
model [4]. Indeed, the nucleon–nucleon coupling via intermediate scattering states
contributes effectively to three-body correlations which cannot be disentangled in
the phenomenological analysis from the correlations generated by the genuine threebody force [1, 2, 5].
Even though the effective interaction theory for nuclear open quantum systems is
not yet completed, nevertheless one can illustrate some of its aspects by analyzing
continuum-coupling correction for the eigenenergy of nuclear closed quantum
system (e.g. the standard shell model), using the real-energy continuum shell model
[5–7]. In this model, the separation of (quasi-)bound (Q subspace) and scattering (P
subspace) states (see Fig. 1.2) is achieved using the projection operator technique
[8]. The total Hamiltonian ˆ
H in Q + P can be formally divided into “unperturbed”
Hamiltonians ˆ
H QQ and ˆ
H PP in Q and P subspaces, respectively, and the coupling
terms ˆ
H QP , ˆ
H PQ between these subspaces. In the Q subspace, the closed quantum
system approximation corresponds to replacing ˆ
H by the standard shell model
Hamiltonian ˆ
H QQ , whereas the open quantum system approximation is described
by an energy-dependent effective Hamiltonian:
ˆ
H QQ (E) = H QQ + H QP G
(+)
P (E) ˆ
H PQ ,
(6.1)
which includes coupling to the scattering continuum. In this expression, G
(+)
P (E)
is a Green’s function for the motion of a single nucleon in P subspace. Hence the
effective Hamiltonian ˆ
H QQ is a complex-symmetric matrix for energies above the
particle-emission threshold (E (thr) ), and Hermitian below it. An eigenvalue
E i (E) = E i + E
(i)
corr
of ˆ
H QQ (E) can be written as a sum of a closed-system eigenenergy E i given by
ˆ
H QQ and the correction due to the coupling to the decay channels. In particular,
this correction depends on the distance of E i from the one-particle threshold and
below the lowest-energy decay threshold (E < 0), i.e. for bound states, the energy
correction is real.
Assuming the coupling to one decay channel, and neglecting the off-diagonal
terms of ˆ
H QP G
(+)
P (E) ˆ
H PQ , the continuum-coupling correction to shell model
energy E i becomes
E
(i)
corr = =Φ i | ˆ
H QP G
(+)
P (E) ˆ
H PQ |Φ i =
∞
0
dr ω i
(+) (r) · w i (r) ,
(6.2)
6 Physical Applications of the Gamow Shell Model
by the monopole components of the two-body shell model Hamiltonian [1, 3]. To
account for this effect in the standard shell model, one would need to introduce
a neutron-number dependence of the T=0 monopole terms [1] which appears if
one includes the three-body interactions in the framework of a standard shell
model [4]. Indeed, the nucleon–nucleon coupling via intermediate scattering states
contributes effectively to three-body correlations which cannot be disentangled in
the phenomenological analysis from the correlations generated by the genuine threebody force [1, 2, 5].
Even though the effective interaction theory for nuclear open quantum systems is
not yet completed, nevertheless one can illustrate some of its aspects by analyzing
continuum-coupling correction for the eigenenergy of nuclear closed quantum
system (e.g. the standard shell model), using the real-energy continuum shell model
[5–7]. In this model, the separation of (quasi-)bound (Q subspace) and scattering (P
subspace) states (see Fig. 1.2) is achieved using the projection operator technique
[8]. The total Hamiltonian ˆ
H in Q + P can be formally divided into “unperturbed”
Hamiltonians ˆ
H QQ and ˆ
H PP in Q and P subspaces, respectively, and the coupling
terms ˆ
H QP , ˆ
H PQ between these subspaces. In the Q subspace, the closed quantum
system approximation corresponds to replacing ˆ
H by the standard shell model
Hamiltonian ˆ
H QQ , whereas the open quantum system approximation is described
by an energy-dependent effective Hamiltonian:
ˆ
H QQ (E) = H QQ + H QP G
(+)
P (E) ˆ
H PQ ,
(6.1)
which includes coupling to the scattering continuum. In this expression, G
(+)
P (E)
is a Green’s function for the motion of a single nucleon in P subspace. Hence the
effective Hamiltonian ˆ
H QQ is a complex-symmetric matrix for energies above the
particle-emission threshold (E (thr) ), and Hermitian below it. An eigenvalue
E i (E) = E i + E
(i)
corr
of ˆ
H QQ (E) can be written as a sum of a closed-system eigenenergy E i given by
ˆ
H QQ and the correction due to the coupling to the decay channels. In particular,
this correction depends on the distance of E i from the one-particle threshold and
below the lowest-energy decay threshold (E < 0), i.e. for bound states, the energy
correction is real.
Assuming the coupling to one decay channel, and neglecting the off-diagonal
terms of ˆ
H QP G
(+)
P (E) ˆ
H PQ , the continuum-coupling correction to shell model
energy E i becomes
E
(i)
corr = =Φ i | ˆ
H QP G
(+)
P (E) ˆ
H PQ |Φ i =
∞
0
dr ω i
(+) (r) · w i (r) ,
(6.2)