References
307
From the |Ψ many-body state, one can build a scattering eigenstate |Ψ (app) of
ˆ
H
(app)
θ
of energy E solution of the following Lippmann–Schwinger equation:
|Ψ
(app)
= |Ψ − ( ˆ
H
(app)
θ
− (E + iη))
−1 ( ˆ
H
(app)
θ
− ˆ
H θ ) |Ψ ,
(6.79)
where η → 0 is a regularizing parameter [164]. One has |Ψ (app) = |Ψ
(a)
res ⊗
|Φ
(A−a)
scat ∀a, as otherwise the equation
E = E res (a) + e
−2iθ E scat (A − a)
holds for a given number of nucleons a. Thus, |Ψ (app) = |Ψ
(a)
scat ⊗ |Φ (A−a)
where |Ψ
(a)
scat is a correlated many-body scattering state of energy E (a) and
|Φ (A−a) is a Slater determinant, with a ∈ 1, 2, . . . , A − 1.
Through the replacements in Eq. (6.79):
A → a, |Ψ → |Ψ
(a)
scat , ˆ
H θ → ˆ
H
(app)
θ
, ˆ
H
(app)
θ
→ ˆ
H
(app’)
θ
,
|Ψ
(app)
→ |Ψ
(a)
scat
(app’) , E → E
(a)
with ˆ
H
(app’)
θ
defined similarly to ˆ
H
(app)
θ
with a → a ∈ 1, 2, . . . , a − 1, one
generates the decomposition |Ψ
(a)
scat
(app’) = |Ψ
(a )
scat (a−a ) , where the energy
of |Ψ
(a)
scat
(app’) ⊗ |Φ (A−a) is equal to E. Similarly to |Ψ (app) |Ψ
(a )
scat
(app’) =
|Ψ
(a )
res ⊗ |Φ
(a−a )
scat
∀a .
Consequently, by recurrence on a , one eventually obtains a many-body state
|Ψ
(1)
scat
(app’) ⊗ |Φ
(A−1)
scat . As |Ψ
(1)
scat
(app’) is a one-body state and |Φ
(A−1)
scat is
a Slater determinant of A − 1 nucleons, |Ψ
(1)
scat
(app’) ⊗ |Φ
(A−1)
scat is a manybody non-interacting state |Φ
(A)
scat of energy E. Therefore, its energy reads
E = e −2iθ E scat (A), which is impossible by assumption. As a consequence,
all scattering eigenstates |Ψ of ˆ
H θ have an energy E of the form E res (a) +
e −2iθ E scat (A − a), with a ∈ 0, 1, 2, . . . , A − 1.
References
1. Y.A. Luo, J. Okołowicz, M. Płoszajczak, N. Michel, arXiv nucl-th/0211068 [nucl-th] (2002)
2. N. Michel, W. Nazarewicz, J. Okołowicz, M. Płoszajczak, J. Rotureau, Acta Phys. Pol. B 35,
1249 (2004)
3. R.J. Charity, K.W. Brown, J. Okołowicz, M. Płoszajczak, J.M. Elson, W. Reviol, L.G.
Sobotka, W.W. Buhro, Z. Chajecki, W.G. Lynch, J. Manfredi, R. Shane, R.H. Showalter, M.B.
Tsang, D. Weisshaar, J.R. Winkelbauer, S. Bedoor, A.H. Wuosmaa, Phys. Rev. C 97, 054318
(2018)
4. A.P. Zuker, Phys. Rev. Lett. 90, 042502 (2003)
5. J. Okołowicz, M. Płoszajczak, I. Rotter, Phys. Rep. 374, 271 (2003)
307
From the |Ψ many-body state, one can build a scattering eigenstate |Ψ (app) of
ˆ
H
(app)
θ
of energy E solution of the following Lippmann–Schwinger equation:
|Ψ
(app)
= |Ψ − ( ˆ
H
(app)
θ
− (E + iη))
−1 ( ˆ
H
(app)
θ
− ˆ
H θ ) |Ψ ,
(6.79)
where η → 0 is a regularizing parameter [164]. One has |Ψ (app) = |Ψ
(a)
res ⊗
|Φ
(A−a)
scat ∀a, as otherwise the equation
E = E res (a) + e
−2iθ E scat (A − a)
holds for a given number of nucleons a. Thus, |Ψ (app) = |Ψ
(a)
scat ⊗ |Φ (A−a)
where |Ψ
(a)
scat is a correlated many-body scattering state of energy E (a) and
|Φ (A−a) is a Slater determinant, with a ∈ 1, 2, . . . , A − 1.
Through the replacements in Eq. (6.79):
A → a, |Ψ → |Ψ
(a)
scat , ˆ
H θ → ˆ
H
(app)
θ
, ˆ
H
(app)
θ
→ ˆ
H
(app’)
θ
,
|Ψ
(app)
→ |Ψ
(a)
scat
(app’) , E → E
(a)
with ˆ
H
(app’)
θ
defined similarly to ˆ
H
(app)
θ
with a → a ∈ 1, 2, . . . , a − 1, one
generates the decomposition |Ψ
(a)
scat
(app’) = |Ψ
(a )
scat (a−a ) , where the energy
of |Ψ
(a)
scat
(app’) ⊗ |Φ (A−a) is equal to E. Similarly to |Ψ (app) |Ψ
(a )
scat
(app’) =
|Ψ
(a )
res ⊗ |Φ
(a−a )
scat
∀a .
Consequently, by recurrence on a , one eventually obtains a many-body state
|Ψ
(1)
scat
(app’) ⊗ |Φ
(A−1)
scat . As |Ψ
(1)
scat
(app’) is a one-body state and |Φ
(A−1)
scat is
a Slater determinant of A − 1 nucleons, |Ψ
(1)
scat
(app’) ⊗ |Φ
(A−1)
scat is a manybody non-interacting state |Φ
(A)
scat of energy E. Therefore, its energy reads
E = e −2iθ E scat (A), which is impossible by assumption. As a consequence,
all scattering eigenstates |Ψ of ˆ
H θ have an energy E of the form E res (a) +
e −2iθ E scat (A − a), with a ∈ 0, 1, 2, . . . , A − 1.
References
1. Y.A. Luo, J. Okołowicz, M. Płoszajczak, N. Michel, arXiv nucl-th/0211068 [nucl-th] (2002)
2. N. Michel, W. Nazarewicz, J. Okołowicz, M. Płoszajczak, J. Rotureau, Acta Phys. Pol. B 35,
1249 (2004)
3. R.J. Charity, K.W. Brown, J. Okołowicz, M. Płoszajczak, J.M. Elson, W. Reviol, L.G.
Sobotka, W.W. Buhro, Z. Chajecki, W.G. Lynch, J. Manfredi, R. Shane, R.H. Showalter, M.B.
Tsang, D. Weisshaar, J.R. Winkelbauer, S. Bedoor, A.H. Wuosmaa, Phys. Rev. C 97, 054318
(2018)
4. A.P. Zuker, Phys. Rev. Lett. 90, 042502 (2003)
5. J. Okołowicz, M. Płoszajczak, I. Rotter, Phys. Rep. 374, 271 (2003)