Transcript "Description of the scattering process using complex-scaling method"

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1
Contents
• Solution of the 5-body Faddeev-Yakubovski equations for
nuclear systems
• Application of the complex-scaling method to solve
scattering problems in Coulombic 3-body systems
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Introduction
Collisions
• In configuration space wave
functions extend to infinity!
• Increasingly complex asymptotic
behaviour for A>2 systems!!
What to do?
Take care of the boundary condition
 Faddeev-Yakubovsky equations
efficiently separates asymptotes of the
binary channels
…
J.W. Waterhouse : « Pandora »
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5-body Faddeev-Yakubovski eq
2
1
4
3
4
2
1
3
3
5
5
5
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4
2
1
4
3
5
1
1
2
4
3
5
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5-body Faddeev-Yakubovski eq
2
1
4
3
2
1
3
4
3
5
5
2
4
2
1
4
3
5
1
1
2
4
5
3
5
NUMERICAL SOLUTION
*R.L., PhD Thesis, Université Joseph Fourier, Grenoble (2003).
• PW decomposition of the components K,H,T,S,F
• Radial parts expanded using Lagrange-mesh method
D. Baye, Physics Reports 565 (2015) 1
• Resulting linear algebra problem solved using iterative methods
• Observables extracted using integral relations
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Numerical costs
2
1
4
3
2
1
3
4
3
5
5
2
2
1
4
4
3
1
4
5
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NUMERICAL SOLUTION
1
2
Problem
Number eq.
(ident particles)
Number eq.
(diff. particles)
PW basis.
Radial
disc.
NN
1
1
2
N
3N
1
3
~100
N2
4N
2
18
~104
N3
5N
5
180
~106
N4
*R.L., PhD Thesis, Université Joseph Fourier, Grenoble (2003).
• PW decomposition of the components K,H,T,S,F
• Radial parts expanded using Lagrange-mesh method
D. Baye, Physics Reports 565 (2015) 1
• Resulting linear algebra problem solved using iterative methods
• Observables extracted using integral relations
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n-4He scattering
MT I-III S-wave potential
120
2
90
P3/2
60

2
P1/2
30
0
-30
No Coulomb
+ Coulomb
-60
2
S1/2
-90
0
4
Ecm (MeV)
8
12
pEFT LO pot.
J. Kircher, PhD Thesis, FA Universitat, Erlangen (2006)
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n-4He scattering
MT I-III S-wave potential
120
2
90
P3/2
60

2
P1/2
30
0
-30
No Coulomb
+ Coulomb
-60
2
Idaho N3LO pot.
S1/2
-90
0
P. Navratil presentation
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Ecm (MeV)
8
12
K.M. Nollett et. al., Phys.Rev.Lett.99:022502,2007
8
n-3H total cross section
2.5
2,5
I-N3LO+3BF(N2LO)
1.5
I-N3LO
MT I-III
Av.18
Av.18+UIX
INOY
1.0
0.5
0.0
0.01
0.1
1
 (b)
 (b)
2.0
2,0
1,5
NLO
Av.18+UIX
INOY
E (MeV)
1,0
0,1
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E (MeV)
9
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Nucleon
Excitation: ~260 MeV
Binding per N: ~8 MeV
p
n
r
a-particle
p
?
h
n
p
Excitation: ~20 MeV
Binding per a: ~3.5 MeV
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Physics of a clusters
Is it possible to parametrize single
pot. for all waves?
Yes
Ali-Bodmer aa potential
L=0
L=2
L=4
V (MeV)
200
But you have to project out the deep-lying
2-body bound states!
100
Buck potential:
0
5.76𝑒𝑟𝑓 0.75 ∗ 𝑥
𝑉 = −122.6225 exp
+
𝑥
S-wave: -72.625 MeV; -25.617 MeV
D-wave: -22.00 MeV
−0.22𝑥2
-100
0
2
4
Raa (fm)
S: 79.1%
D: 19.8 %
G: 1.13%
Problem
Ali-Bodmer
Buck
PappMoszkowski
(fish-bone)*
Exp.
(MeV)
8Be
0.0828
0.179
0.0918
0.0918
12C
-1.52
-0.28
-7.27
-7.275
16O
-36.3
-57.0
-14.4
(0+)
aa
S: 7.5%
D: 24.0 %
G: 64.9%
(0+)
aaa
(0+)
aaaa
*Mod.Phys.Lett.B22:2201-2215,2008
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Introduction
Collisions
• In configuration space wave
functions extend to infinity!
• Increasingly complex assymptotic
behavior for A>2 systems!!
What to do?
Take care of the boundary condition
 Faddeev-Yakubovsky equations
efficiently separates assymptotes of the
binary channels
BUT
 Number of the reaction channels increases
rapidly with A
…
 Even for 3-body systems there may exist
infinite number of binary channels
e+He+H(n=1,2,…,)
 Complex behavior of the breakup
assymptotes
FIND SOME TRICKS TO AVOID
PROBLEMS AT BOUNDARY!
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Complex scaling: resonances, reactions & bound states in an unified formalism
In Quantum-Mechanics problem of N-particle dynamics may be often
formulated in time-independent formalism and for the wave functions
whose asymptotes contain only nontrivial outgoing waves
𝐸𝑠𝑐 − 𝐻0 −
𝑉𝑖𝑗 (𝑟𝑖𝑗 ) 𝜓 𝒓𝒊 = 𝑆 𝒓𝒊
𝑘𝐼
𝑖>𝑗
•
Bound state problem: 𝑆 𝒓𝒊 ≡ 𝟎
B. States
Scattering
•
Resonances: 𝑆 𝒓𝒊 ≡ 𝟎
•
Reactions due to external probe: 𝑆 𝜌 → ∞ = 𝟎
𝑘𝑅
𝜓(𝜌 → ∞) ∝ 𝑒 𝑖𝑘𝑥 𝑟𝑥
•
Collisions: 𝑆 𝒓𝒊 ≡ 𝟎
𝐴𝑐 (𝑘𝑐 ) 𝑒 𝑖|𝑘𝑐 |𝑟𝑐
𝜓𝑠𝑐 (𝜌 → ∞) ∝ 𝜓𝑖𝑛 𝒓𝒊 +
𝑐
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Resonances
𝑘𝑟𝑒𝑠 = 𝑘𝑅 − 𝑖𝑘𝐼 = |𝑘𝑟𝑒𝑠 |𝑒 −𝑖𝜂𝑟𝑒𝑠
Complex scaling: resonances, reactions & bound states in an unified formalism
R. Hartree , J.G. L. Michel, P. Nicolson (1946)
J. Nuttal and H. L. Cohen, Phys. Rev. 188 (1969) 1542
Complex scaling kills outgoing waves in the assymptote
r
 re i
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𝜓 𝜌 → ∞ ∝ 𝑒 𝑖𝑘𝑥 𝑟𝑥 𝑒
𝑖𝜃
= 𝑒 𝑖𝑘𝑥 𝑟𝑥 𝑐𝑜𝑠𝜃 𝑒 −𝑘𝑥 𝑟𝑥 𝑠𝑖𝑛𝜃
Complex scaling: resonances, reactions & bound states in an unified formalism
In Quantum-Mechanics problem of N-particle dynamics may be often
formulated in time-independent formalism and for the wave functions
whose asymptotes contain only nontrivial outgoing waves
𝐸𝑠𝑐 − 𝐻0 −
𝑉𝑖𝑗 (𝑟𝑖𝑗 ) 𝜓 𝒓𝒊 = 𝑆 𝒓𝒊
𝑖>𝑗
•
Bound state problem: 𝑆 𝒓𝒊 ≡ 𝟎
•
Resonances: 𝑆 𝒓𝒊 ≡ 𝟎
•
Reactions due to external probe: 𝑆 𝜌 → ∞ = 𝟎
𝜓(𝜌 → ∞) ∝ 𝑒 𝑖𝑘𝑥 𝑟𝑥
•
Collisions: 𝑆 𝒓𝒊 ≡ 𝟎
𝐴𝑐 (𝑘𝑐 ) 𝑒 𝑖|𝑘𝑐 |𝑟𝑐
𝜓𝑠𝑐 (𝜌 → ∞) ∝ 𝜓𝑖𝑛 𝒓𝒊 +
Diverges
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𝑐
Complex scaling: resonances, reactions & bound states in an unified formalism
R. Hartree , J.G. L. Michel, P. Nicolson (1946)
J. Nuttal and H. L. Cohen, Phys. Rev. 188 (1969) 1542
Complex scaling kills outgoing waves in the assymptote
r
 re i
•
𝜓 𝜌 → ∞ ∝ 𝑒 𝑖𝑘𝑥 𝑟𝑥 𝑒
𝑖𝜃
exp. bound if 0<<p
= 𝑒 𝑖𝑘𝑥 𝑟𝑥 𝑐𝑜𝑠𝜃 𝑒 −𝑘𝑥 𝑟𝑥 𝑠𝑖𝑛𝜃
Schrödinger equation in a driven form:
𝑟Ψ 𝑟 = 𝐹𝑙
ℏ2
2𝜇
𝑑2
− 2
𝑑𝑟
+
𝑖𝑛
𝑟 + 𝐹𝑙
𝑙 𝑙+1
𝑟2
𝑑2
− 𝑒 2𝑖𝜃𝑑𝑟 2
ℏ2
𝑑2 𝑙 𝑙 + 1
2 𝐹
− 2+
−
𝑘
𝑙
2𝜇
𝑑𝑟
𝑟2
(𝑟)
+ 𝑉𝑙 𝑟 − 𝑘 2 𝐹𝑙
𝑠𝑐
𝑟 =-𝑉𝑙 𝑟 𝐹𝑙
𝑖𝑛
𝑖𝑛
𝑟
𝑙 𝑙+1
+ 𝑒 2𝑖𝜃𝑟 2 + 𝑉𝑙 𝑟𝑒 𝑖𝜃 − 𝑘 2 𝐹𝑙
𝑠𝑐
𝑟 =-𝑉𝑙 𝑟𝑒 𝑖𝜃 𝐹𝑙
𝑖𝑛
𝑟𝑒 𝑖𝜃
exp. bound for the short
range pot. term Vs
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𝑟 =0
r
 re i
Complex scaling
ℏ2
2𝜇
𝑠𝑐
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Complex scaling: resonances, reactions & bound states in an unified formalism
Presence of Coulomb (or long range interaction)
𝑉𝑙 𝑟 = 𝑉𝑙 𝑆 𝑟 +𝑉𝑙
𝑙𝑜𝑛𝑔
𝑟
ℏ2
𝑑2 𝑙 𝑙 + 1
2 + 𝑉 𝑙𝑜𝑛𝑔 (𝑟) 𝐹
− 2+
−
𝑘
𝑙
𝑙
2𝜇
𝑑𝑟
𝑟2
•
𝑖𝑛
𝑟 =0
Schrödinger equation in a driven form:
𝑟Ψ 𝑟 = 𝐹𝑙
ℏ2
2𝜇
𝑑2
− 𝑑𝑟 2
+
𝑖𝑛
𝑟 + 𝐹𝑙
𝑙 𝑙+1
𝑟2
𝑑2
− 𝑒 2𝑖𝜃𝑑𝑟 2
(𝑟)
+ 𝑉𝑙 𝑟 − 𝑘 2 𝐹𝑙
𝑠𝑐
𝑟 =-𝑉𝑙 𝑆 𝑟 𝐹𝑙
𝑖𝑛
𝑟
𝑉𝑙 𝑟 − 𝑉𝑙
r
 re i
Complex scaling
ℏ2
2𝜇
𝑠𝑐
𝑙 𝑙+1
+ 𝑒 2𝑖𝜃𝑟 2 + 𝑉𝑙 𝑟𝑒 𝑖𝜃 − 𝑘 2 𝐹𝑙
𝑠𝑐
𝑟 =- 𝑉𝑙 𝑆 𝑟𝑒 𝑖𝜃 𝐹𝑙
𝑖𝑛
𝑙𝑜𝑛𝑔
𝑟𝑒 𝑖𝜃
exp. bound for the short
range pot. term Vs
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Complex scaling: reactions
Subtelties for N>2 case A. Deltuva, R.L. et al.,PPNP 74 (2014)
• Complex scaling parameter is limited
• Incorporation of long range interactions
Inhomogenious term in the driven Schrödinger eq:
𝑎
𝑟𝑎,𝑏
𝑏
1.0
𝑁
𝑉𝑖𝑗 𝑟𝑖𝑗 − 𝑉 𝑖𝑗𝐶(𝑟𝑎,𝑏 )
𝑓𝑐𝑢𝑡(𝑟𝑎,𝑏 )
𝑖∈𝑎;𝑗∈𝑏
 Long-range residual terms ~
𝜓𝑏𝑠 𝑟𝑎 , 𝑟𝑏 𝐹𝑙
𝑖𝑛(𝑞
𝑎,𝑏 𝑟𝑎,𝑏 )
fcut(ra,b)
Physical
region ra,b~1
0.5
0.0
1
(𝑟𝑎,𝑏 )𝑛
0
with n≥2 for problems including Coulomb
 For the problem dominated by short-range interactions these residual terms are not
important! Clearly we may screen this term!
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2
ra,b
4
Review of the method*
•
Complex scaling method is very functional, adaptable to almost any b.s. technique.
Already successfully applied using:
 Spline basis
 Laguerre, HO, Gaussian, correlated Gaussian, Sturmian basis functions
 Lagrange mesh method
T. Myo, Y. Kikuchi, and K. Katō, Phys. Rev. C 85, 034338 (2012) , Attila Csótó, Phys. Rev. C 49, 2244 (1994), B. Gyarmati
and A. T. Kruppa, Phys. Rev. C 34, 95 (1986)
•
External probes

2-body, 3-body,4-body systems, including repulsive Coulomb
T.M, K. Kato, S. Aoyama and K. Ikeda PRC63(2001)054313, T.Myo, K. Kato, H. Toki, K. Ikeda, PRC76(2007) 024305
•
Collisions, demonstrated to work for:






1
2-body collisions including Coulomb interaction, Optical potentials, 𝑛 potentials with n≥4
𝑟
3-body scattering including the break-up
3-body scattering with Optical potential
3-body scattering for the systems, where two-particles (clusters) are charged
3-body break-up amplitude for n-d & p-d scattering
4-body scattering in 4N systems, including Coulomb
…
R.L. & J. Carbonell, Phys. Rev. C 84, 034002 (2011); A. Deltuva, R.L., A.C. Fonseca, “Clusters in Nuclei Vol.3 –LNP 875, 1
(2013); A. Deltuva, R.L. et al.,PPNP 74 (2014) ; R.L., Phys. Rev. C 91, 041001(R) (2015),..
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And at extreme (Coulombic systems)??
What if all particles are charged, purely Coulombic system?
 Exterior complex scaling method
• Workable for 3-body sytem, however success is limited
M.V. Volkov et al.: EPL, 110 (2015) 30006, C.W. McCurdy, PRA 63, 022711
• Difficult/impossible to extend beyond n=3
• In conflict with PW decomposition
 Can we make smooth CS to work in Coulombic systems?

r
And at extreme (Coulombic systems)??
1
1
1
𝑥2
𝑦1
3
𝑦2
L.D. Faddeev: Zh. Eksp. Teor. Fiz. 39, (1960. 1459
[Sov. Phys. JETP 12, (1961) 1014.]
𝐹𝛼 𝑥𝛼 , 𝑦𝛼 = 𝐺0 𝑉𝛼 𝑥𝛼 Ψ
for short range potentials separate
assymptotes of the binary scattering
channels
3
𝑥1
2
2
2
𝑦3
𝑥3
Faddeev equation:
(𝐸 − 𝐻0 − 𝑉𝛼 𝑥𝛼 )𝐹𝛼 𝑥𝛼 , 𝑦𝛼 = 𝑉𝛼 𝑥𝛼 𝐹𝛽 𝑥𝛽 , 𝑦𝛽 + 𝐹𝛾 𝑥𝛾 , 𝑦𝛾
does not workt for Coulomb interaction
Generalized by Merkuriev :
S.P. Merkuriev: Ann. Phys. (N.Y.) 130 (1980) 395
•
Potential is split into two parts
𝑉𝛼 𝑥𝛼 = 𝑉𝛼 𝑙𝑜𝑛𝑔 𝑥𝛼 , 𝑦𝛼 + 𝑉𝛼 𝑠ℎ𝑜𝑟𝑡 𝑥𝛼 , 𝑦𝛼
•
Long range parts
forms 3-body potential, which is added to H0
3
𝑉𝑖 𝑙𝑜𝑛𝑔 −𝑉𝛼 𝑠ℎ𝑜𝑟𝑡)𝐹𝛼 𝑥𝛼 , 𝑦𝛼 = −𝑉𝛼 𝑠ℎ𝑜𝑟𝑡 𝐹𝛽 𝑥𝛽 , 𝑦𝛽 + 𝐹𝛾 𝑥𝛾 , 𝑦𝛾
(𝐸 − 𝐻0 −
𝑖=1
And at extreme (Coulombic systems)??
3
𝑉𝑖 𝑙𝑜𝑛𝑔 −𝑉𝛼 𝑠ℎ𝑜𝑟𝑡)𝐹𝛼 𝑥𝛼 , 𝑦𝛼 = −𝑉𝛼 𝑠ℎ𝑜𝑟𝑡 𝐹𝛽 𝑥𝛽 , 𝑦𝛽 + 𝐹𝛾 𝑥𝛾 , 𝑦𝛾
(𝐸 − 𝐻0 −
𝑖=1
Solution with smooth CS:
•
We separate incomming wave
𝐹𝛼 𝑖𝑛 𝑥𝛼 , 𝑦𝛼 + 𝐹𝛼 𝑠𝑐 𝑥𝛼 , 𝑦𝛼
with
𝐹𝛼 𝑥𝛼 , 𝑦𝛼 =
𝐸 − 𝐻0 − 𝑉𝛼 𝑅 −𝑉𝛼 𝑠ℎ𝑜𝑟𝑡 𝐹𝛼 𝑖𝑛 𝑥𝛼 , 𝑦𝛼 = 0
3
where 𝑉𝛼 𝑅 tries to approximate assymptotics of
𝑉𝑖 𝑙𝑜𝑛𝑔
𝑖=1≠𝛼
There remains one troubling term, before performing smooth CS:
1.0
Physical
region ra,b~1
𝑉𝑖 𝑙𝑜𝑛𝑔 𝐹𝛼 𝑖𝑛 𝑥𝛼 , 𝑦𝛼
𝑓𝑐𝑢𝑡(𝑦𝛼 ) 𝑉𝛼 𝑅 −
fcut(ra,b)
3
0.5
𝑖=1≠𝛼
0.0
0
2
ra,b
4
Numerical solution
Solution: FY equations=PW decomposition
+ Lagrange-mesh discretization of the radial parts
+Iterative methods to solve linear algebra problem
D. Baye, Phys. Stat. Sol. B 243 (2006) 1095
+Khon variational principle to extract scattering amplitudes
C. Romero-Redondo et al., Phys. Rev. A 83, 022705 (2011)
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e-Ps scattering
600
35
Total (l=S,P,D,F,G)
S
P
D
30
25
st
2
el (pa0)
2
r (pa0)
400
Total (l=S,P,D,F,G)
S
P
D
20
ionisation
Inelastic e-Ps(1s) cross section
1 excitation
Elastic e-Ps(1s) cross section
15
200
10
5
0
0.0
0.1
0.2
0.3
Ecm (a.u.)
R. Lazauskas
0.4
0.5
0
0.0
0.1
0.2
0.3
Ecm (a.u.)
0.4
0.5
e-H scattering
2
r (pa0)
2
3
ionisation
30
el (pa0)
4
Total (l=S,P,D,F,G)
S
P
D
st
40
1 excitation
Inelastic e-H(1s) cross section
Elastic e-H(1s) cross section
20
10
0
0.2
0.3
0.4 0.5 0.6
Ecm (a.u.)
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0.7
0.8
Total (l=S,P,D,F,G)
S
P
D
2
1
0
0.3
0.4
0.5
0.6
Ecm (a.u.)
0.7
0.8
e+-H(1s)
p – Ps(1s)
e+-H(1s)
p – Ps(1s)
e+-H(1s) cross section
p - PS(1s) cross section
200
45
p-PS(1s) cross sections
elastic
non-elastic
H(1s) production
30
2
 (pa0)
150
2
 (pa0)
e+ -H/p-Ps scattering
+
e -H(1s) cross sections
elastic
non-elastic
PS(1s) production
15
0
0.3
0.4
0.6
0.5
Ecm (a.u.)
R. Lazauskas
0.7
100
50
0
0.0
0.1
0.2
0.3
Ecm (a.u.)
0.4
0.5
e+ -H/p-Ps scattering
e+-H(1s)
p – Ps(1s)
H-prod (a.u.)
50
40
30
20
free wave
upto (n=2)
4
(n=2)+a/r
4
(n=3)+a/r
10
0
0.00
0.05
E (a.u.)
R. Lazauskas
0.10
Conclusion
• In a few last years several different methods have been developed enabling to
solve one of the most challenging few-body problems, related with many-body
breakup
• In particular, complex-scaling method has been revived in nuclear physics.
This method enables to solve few-body scattering problem employing standard
bound state techniques (feasible by almost any config. space bound state
technique and requires very limited effort to be implemented)
• Simple extension of the formalism to many-body scattering case
• Very accurate results are already obtained for 3-body and 4-body elastic and
breakup scattering
• Solution of the Coulombic problems is possible
Acknowledgements: The numerical calculations have been performed at IDRIS (CNRS, France). We
thank the staff members of the IDRIS computer center for their constant help.
19/07/2016
R. Lazauskas
29
Complex scaling method, collisions
• Solution (using standard bound state techniques)
1.
Expand c.s. outgoing wave in your favorite basis
Complex coefficients
𝑁
Ψ 𝑠𝑐 𝑟 ≈
𝑐𝑖 𝜓𝑖 (𝑟)
square integrable basis functions
𝑖=1
2.
Convert c.s. Schrödinger equation into linear algebra problem
𝜓𝑗 𝑟 𝑑𝑟 ⋮
𝑗 = 1, . . 𝑁

d2
l (l  1)
i
2
sc
s
i
in
i

 2i 2  2i 2  Vl (re )  k  l (r )  Vl (re )l (re )
2  e dr
e r

2
𝐻𝑗,𝑖 𝑐𝑖 = 𝑏𝑗
3.
4.
Solve linear algebra problem to determine coefficients 𝑐𝑖
i.
Spectral expansion (Y.Suzuki, W.Horiuchi, K. Kato): determine all
eigenvalues/eigenvectors
ii. (Iterative) linear algebra methods
Extract scattering observables from obtained solution(s)
19/07/2016
R. Lazauskas