296
6 Physical Applications of the Gamow Shell Model
B. Write the many-body operator ˆ
H θ for the case of non-interacting particles.
Deduce that the eigenenergies of ˆ
H θ can be written as
E res (a) + e
−2iθ E scat (A − a),
where a = 0, 1, 2, . . . , A, E res (a) is the energy of a resonant state of a
nucleons (E res (0) = 0 by convention), and E scat (A − a) ≥ 0. Demonstrate
that the associated ˆ
H θ eigenstates of ˆ
H θ form a complete set of states in the
Hilbert space.
C. Show that the eigenenergies formulas of B are still valid when particles interact
via a short-range force. For this, one will consider separately resonant and
scattering eigenstates. Show that the case of resonant eigenstates can be treated
as in the Gamow shell model. For scattering eigenstates, one will consider
Hamiltonians consisting of several non-interacting parts and determine their
eigenenergies E. The scattering eigenenergy of ˆ
H θ will be shown to be equal
to E using the Lippmann–Schwinger equation. This approach is inspired by
that developed in Ref. [140].
Note that potentials and interactions do not have to be dilation analytic in the
Gamow shell model. Indeed, exterior complex scaling is used in the Gamow shell
model to calculate matrix elements (see Sect. 3.3), i.e. complex scaling is used only
in the asymptotic zone. As a consequence, dilation analyticity is trivially obtained,
because only centrifugal and Coulomb potentials subsist at large distances.
6.6.2 Basis Dependence in Truncated Model Spaces
The theoretical results presented in Sect. 6.6.1 provide a simple and well defined
method to determine the eigenstates of ˆ
H . For this, it is sufficient to diagonalize the
complex-scaled operator ˆ
H θ (see Eq. (6.67)) with a basis of bound states using a
sufficiently large θ angle. However, in practice, it is necessary to truncate the basis,
so that one has to assess the rate of convergence obtained with the number of basis
states.
6.6.2.1 Basis Optimization
The fundamental problem is to precisely reproduce the many-body wave function
in the asymptotic region. Let us firstly consider the case of halo states to ponder out
this problem. It is straightforward to verify from Eq. (6.66) that the exponential
decrease of weakly bound one-body states is slow ∀θ < π/4. Consequently,
one faces the same problem with the diagonalization of initial Hamiltonian or
Hamiltonian transformed with complex scaling. Indeed, the most immediate choice
for the basis of bound states is that of a basis of harmonic oscillator states, with
which convergence when expanding weakly bound states is very slow due to the
Gaussian fall-off of harmonic oscillator states. The situation is the same if one
aims at calculating resonance states. Indeed, even though the application of the ˆ
U θ
6 Physical Applications of the Gamow Shell Model
B. Write the many-body operator ˆ
H θ for the case of non-interacting particles.
Deduce that the eigenenergies of ˆ
H θ can be written as
E res (a) + e
−2iθ E scat (A − a),
where a = 0, 1, 2, . . . , A, E res (a) is the energy of a resonant state of a
nucleons (E res (0) = 0 by convention), and E scat (A − a) ≥ 0. Demonstrate
that the associated ˆ
H θ eigenstates of ˆ
H θ form a complete set of states in the
Hilbert space.
C. Show that the eigenenergies formulas of B are still valid when particles interact
via a short-range force. For this, one will consider separately resonant and
scattering eigenstates. Show that the case of resonant eigenstates can be treated
as in the Gamow shell model. For scattering eigenstates, one will consider
Hamiltonians consisting of several non-interacting parts and determine their
eigenenergies E. The scattering eigenenergy of ˆ
H θ will be shown to be equal
to E using the Lippmann–Schwinger equation. This approach is inspired by
that developed in Ref. [140].
Note that potentials and interactions do not have to be dilation analytic in the
Gamow shell model. Indeed, exterior complex scaling is used in the Gamow shell
model to calculate matrix elements (see Sect. 3.3), i.e. complex scaling is used only
in the asymptotic zone. As a consequence, dilation analyticity is trivially obtained,
because only centrifugal and Coulomb potentials subsist at large distances.
6.6.2 Basis Dependence in Truncated Model Spaces
The theoretical results presented in Sect. 6.6.1 provide a simple and well defined
method to determine the eigenstates of ˆ
H . For this, it is sufficient to diagonalize the
complex-scaled operator ˆ
H θ (see Eq. (6.67)) with a basis of bound states using a
sufficiently large θ angle. However, in practice, it is necessary to truncate the basis,
so that one has to assess the rate of convergence obtained with the number of basis
states.
6.6.2.1 Basis Optimization
The fundamental problem is to precisely reproduce the many-body wave function
in the asymptotic region. Let us firstly consider the case of halo states to ponder out
this problem. It is straightforward to verify from Eq. (6.66) that the exponential
decrease of weakly bound one-body states is slow ∀θ < π/4. Consequently,
one faces the same problem with the diagonalization of initial Hamiltonian or
Hamiltonian transformed with complex scaling. Indeed, the most immediate choice
for the basis of bound states is that of a basis of harmonic oscillator states, with
which convergence when expanding weakly bound states is very slow due to the
Gaussian fall-off of harmonic oscillator states. The situation is the same if one
aims at calculating resonance states. Indeed, even though the application of the ˆ
U θ