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Two-body Coulomb sca e ing and complex scaling
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2013 J. Phys.: Con . Se . 436 012032
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Two-body Coulomb sca e ing and complex scaling
Is ´an Ho ny´ak
Uni e si y o Deb ecen, Facul y o In o ma ics, PO Box 12, 4010 Deb ecen, Hunga y
E-mail: [email p o ec ed]
Abs ac . The wo-body Coulomb sca e ing p oblem is s udied wi h he complex scaling
me hod. Ou spli ing me hod [1], on pa ial wa e le el, simpli ies he sca e ing bounda y
condi ion. The sca e ed pa o he wa e unc ion is squa e in eg able. This p ope y makes
he nume ical solu ion easie .
1. In oduc ion
The me hod o complex scaling (CS) has been success ully applied in many a eas o quan um
physics, bu o sca e ing p oblems he CS p ocedu e can only be applied o sho - ange
po en ials, which limi s he applicabili y o CS in a omic and nuclea physics.
I has been shown in he Re . [2] ha sca e ing calcula ions wi h he ex e io CS can be
success ully pe o med o long- ange in e ac ions as well. Ne e heless, he ex e io CS me hod
has been unde sc u iny since an a i icial cu o o some o he in e ac ion may cause oubles.
Recen ly, howe e , i has been shown [3] ha he s anda d CS can be applied o sca e ing
p oblems e en in he p esence o he pu e Coulomb in e ac ion. The me hod is based on he
wo-po en ial o malism. An al e na i e echnique is p esen ed below.
The ull sca e ing solu ion is w i en in he o m
ψ+(k, ) = φ0(k, ) + ψsc+(k, ),(1)
whe e ψsc+(k, ) is he sca e ed wa e, and φ0(k, ) is a known unc ion. F om he Sch ¨odinge
equa ion he so called d i en Sch ¨odinge equa ion can be de i ed o he sca e ed wa e:
(E−ˆ
H)ψsc+(k, ) = S(k, ),(2)
whe e he sou ce e m is gi en by S(k, ) = ( ˆ
H−E)φ0(k, ).
2. Theo y
We ha e ound ha he choice o he so called Coulomb-modi ied plane wa e (CMPW),
φ0(k, ) = eik (k −k )iγ,(3)
does no simpli y he bounda y condi ion om he poin o iew o he complex scaling.
We a he spli he wa e unc ion in o incoming and sca e ed wa es a he pa ial-wa e le el.
The exac incoming wa e is ψi,l(k, ) [4] and we w i e
ψ+
l(k, ) = ψi,l(k, ) + ψsc+
l(k, ),(4)
10 h In e na ional Con e ence on Clus e ing Aspec s o Nuclea S uc u e and Dynamics IOP Publishing
Jou nal o Physics: Con e ence Se ies 436 (2013) 012032 doi:10.1088/1742-6596/436/1/012032
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whe e
ψi,l(k, ) = ωi,l(k, ) + χl(k, ) (5)
and
ωi,l(k, ) = e−ik eγπ/2(−1)l+1(2ik )lU(l+ 1 −iγ, 2l+ 2,2ik ),(6)
χl(k, ) = eik +γπ/2
2ik
(−1)l
(2ik )l
Γ(2l+ 1)
Γ(l+ 1 −iγ)
l
X
n=0
(−1)n(iγ −l)n
(−2l)nn!(2ik )n,(7)
wi h Ubeing he con luen hype geome ic unc ion. Fo he d i en adial Sch ¨odinge equa ion
di ec calcula ion gi es he ollowing sou ce e m
Sl(k, ) = eik +γπ/2
2 2Γ(−iγ).(8)
A e CS by scaling angle θ, which sa is ies he condi ion 0 < θ < π, and he ans o ma ion
hsc+
l,θ (k, ) = l+1ψsc+
l,θ (k, ), which esul s in a egula unc ion hsc+
l,θ (k, ) a = 0) we ge he
d i en adial Sch ¨odinge equa ion
"k2
2+e−2iθ 1
2
d2
d 2−e−2iθ l
d
d −e−iθ γk
−Vs( eiθ)#hsc+
l,θ (k, ) = l+1S o
l,θ (k, ),(9)
whe e he new sou ce e m eads
S o
l,θ (k, ) = Sl,θ(k, ) + ei3θ/2ψi,l(k, eiθ)Vs( eiθ).(10)
The bounda y condi ions a e
hsc+
l,θ (k, 0) = 0 (11)
and
lim
→∞ hsc+
l,θ (k, )=0.(12)
One can show ha he phase shi δlcan be exp essed as he → ∞ limi o a unc ion σl( ),
which we call he local ep esen a ion o he phase shi :
e2iσl( )= (2k eiθ)iγ "e−ik eiθ 2ik e−iθ/2˜
ψsc+
l,θ (k, ) + eγπ/2(−1)l(iγ −l)l
Γ(l+ 1 −iγ)#→e2δl( → ∞).(13)
3. Nume ical Resul s
In nume ical calcula ions he bounda y condi ion (12) is app oxima ed by a simila exp ession
o a la ge bu ini e alue Ro , which in in he asymp o ic egion:
hsc+
l,θ (k, R)=0.(14)
The ini e elemen me hod is used as a nume ical echnique o he solu ion o Eq. (9). In
he calcula ions equally spaced ini e elemen s o leng h 1 a omic uni a e aken. The deg ee o
he Loba o shape unc ions is deno ed by N, and he same N alue is chosen o each elemen .
A sho - ange po en ial Vs= 7.5 2exp(− ) is added o he Coulomb po en ial.
In Figu e 1 a case o θ= 0.1 adian is shown, wi h he angula momen um l= 0 and k= 3.
The Coulomb po en ial is gi en by 1/ . The local ep esen a ion o he phase shi is p ac ically
cons an in a huge egion, which can be iden i ied wi h an app oxima e phase-shi alue. In
con as he local app oxima ion o he phase shi in [5] ends o he exac alue by a pe sis en
oscilla ion wi h dec easing o de o ampli ude.
10 h In e na ional Con e ence on Clus e ing Aspec s o Nuclea S uc u e and Dynamics IOP Publishing
Jou nal o Physics: Con e ence Se ies 436 (2013) 012032 doi:10.1088/1742-6596/436/1/012032
2
200 400 600 800 1000
-0.18
-0.178
-0.176
phase shi
exac
R=250
R=1000
200 400 600 800 1000
0.9
0.92
0.94
0.96
phase shi
Coulomb
Coulomb + sho ange
Figu e 1. The local ep esen a ion o he phase shi o he pu e Coulomb po en ial (uppe
pa ) and o he Coulomb plus sho - ange po en ial (lowe pa ). Fo he pu e Coulomb case
he dashed line displays he exac solu ion.
4. Conclusion
The wo-body sca e ing p oblem has been sol ed o a pu e Coulomb po en ial as well as o
a Coulomb plus ini e- ange po en ial using he s anda d complex scaling me hod. No cu o
is applied o he ail o he long- ange in e ac ion. The CS makes i possible o obse e he
sca e ing bounda y condi ion wi h g ea accu acy in a ai ly simple manne .
Re e ence
[1] Ho nyak I and K uppa A T 2012 Phys. Re . A85 022702
[2] Rescigno T N, Bae schy M, By um D and McCu dy C W 1997 Phys. Re . A55 4253
[3] K uppa A T, Suzuki R and Ka o K 2007 Phys. Re . C75 044602
[4] an Hae ingen H 1976 Il Nuo o Cimen o 34B 53
[5] Volko M V, Elande N, Ya e sky E and Yako le S L 2009 Eu ophys. Le . 85 30001
10 h In e na ional Con e ence on Clus e ing Aspec s o Nuclea S uc u e and Dynamics IOP Publishing
Jou nal o Physics: Con e ence Se ies 436 (2013) 012032 doi:10.1088/1742-6596/436/1/012032
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