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Unified Description of Bound and Unbound States -- Resolution of Identity --. KEK Lecture (2). K. Kato Hokkaido University Oct. 6, 2010. 1. Resolution of Identity in the Complex Scaling Method. Completeness Relation (Resolution of Identity). R.G. Newton, J. Math. Phys. 1 (1960), 319. - PowerPoint PPT Presentation
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Unified Description of Bound and Unbound States
-- Resolution of Identity --
K. Kato
Hokkaido University
Oct. 6, 2010
KEK Lecture (2)
1. Resolution of Identity in the Complex Scaling Method
Continuum states Bound statesResonant states
non-resonant continuum states
|~|~|1 1kkRnn
bn
dkuu
L
R.G. Newton, J. Math. Phys. 1 (1960), 319
Completeness Relation (Resolution of Identity)
Among the continuum states, resonant states are considered as an extension of bound states because they result from correlations and interactions.
Separation of resonant states from continuum states
|~|~|~|1 1)(
kkL
LN
rnrrnn
bn
dkuuuur
Deformation of the contour
Resonant states
ˆ~ lim ˆ~2
*1
021
2
uOuedruOu r
R
Ya.B. Zel’dovich, Sov. Phys. JETP 12, 542 (1961).
N. Hokkyo, Prog. Theor. Phys. 33, 1116 (1965).
Convergence Factor Method
Matrix elements of resonant states
T. Berggren, Nucl. Phys. A 109, 265 (1968)
Deformed continuum states
Complex scaling method
irer
ikek
coordinate:
momentum:
r
)()(ˆ)(~ )(
ˆ~ lim ˆ~
2*
1
2*
10
21
2
ii
R
i
r
R
reuOreured
uOuedruOu
|~|~|~|1 1
kkL
N
rnnnnn
bnk
r
dkuuuu
B. Gyarmati and T. Vertse, Nucl. Phys. A160, 523 (1971).
reiθ
T. Myo, A. Ohnishi and K. Kato. Prog. Theor. Phys. 99(1998)801]Rotated Continuum states Resonant states
J. Aguilar and J.M.Combes, J. Math. Phys. 22, 269 (1971)
E.Balslev and J.M.Combes, J. Math. Phys. 22, 280 (1971)
k k E E
Single Channel system B.Giraud and K.Kato, Ann.of Phys. 308 (2003), 115.
Resolution of Identity in Complex Scaling Method
0 0
Eigenvalues of H(θ) with a L2 basis set
(L2 basis set ; Gaussian basis functions)
b1b2b3 r1r2 r3
Coupled Channel system Three-body system
E| E|
B.Giraud, K.Kato and A. Ohnishi, J. of Phys. A37 (2004),11575
0
Bany-body system
|~|"|~|'|~|~|~|1 ""1
''11
"'
kkLkkLkkL
N
rnnnnn
bnkkk
r
dkdkdkuuuu
9Li+n+n 10Li(1+)+n 10Li(2+)+n Resonances
T. Myo, A. Ohnishi and K. Kato, Prog. Theor. Phys. 99 (1998), 801.
in CSMin CSM
10Li :9Li(3/2-)
+n(p3/2)
Complex Scaled Green’s Functions
Green’s operator
)(
1)(
HEG
|~|~|~|1 1
kkL
N
rnnnnn
bnk
r
dkuuuu
Resolution of Identity
Complex Scaled Green’s function
Complex scaled Green’s operator
iHEG
1)(
2. Strength Functions and Coulomb Breakup Reaction
Coulomb Breakup Reactions of Three-Body Systems (two-neutron halo systems)
Breakup Mechanism: Direct or Sequential ?
Simultaneous description of Structure and Reaction
Neutron-correlation in 2-neutron halo states
T. Myo, K. Kato, S. Aoyama and K. Ikeda, PRC63(2001), 054313
Coulomb breakup strength of 6He( , )
( )EE
d dB E EN E
dE dE
( ) : virtual photon numberEN E
6He : 240MeV/A, Pb Target (T. Aumann et.al, PRC59(1999)1252)
T.Myo, K. Kato, S. Aoyama and K. IkedaPRC63(2001)054313.
Coulomb breakup cross section of 11Li
T. Nakamura et al., Phys. Rev. Lett. 96, 252502 (2006)
)()( iEEE
A.T.Kruppa, Phys. Lett. B 431 (1998), 237-241
A.T. Kruppa and K. Arai, Phys. Rev. A59 (1999), 2556
K. Arai and A.T. Kruppa, Phys. Rev. C 60 (1999) 064315
Definition of LD:
iHETrE
1Im
1)(
iii EH
3. Continuum level density
B
B
B
R RB
N
n
N
nL C
CRn
Bn EE
dEEEEE
iHETrE
111Im
1
1Im
1)(
1
Resonance:
Continuum:2R
RR
nRn
Rn iE
IRC iE
LIR
ICN
n nRn
nN
n
Bn E
dEE
EER
R RR
RB
B
B 2222 )(
1
4/)(
2/1)(
Discreet distribution
RI in complex
scaling
2θ
2θ
E E
εI εI
Discretized Continuum States in the Complex Scaling Method
Continuum Level Density: )()()( 0 EEE
)()(Im1
11Im
1)(
0
0
EGEGTr
iHEiHETrE
Basis function method: n
N
nnc
1
Phase shift calculation in the complex scaled basis function method
)()(
2
1)( ES
dE
dESTr
iE
In a single channel case, )}(2exp{)( EiES
dE
EdE
)(1)(
)'(')(0
EdEEE
S.Shlomo, Nucl. Phys. A539 (1992), 17.
Phase shift of 8Be=+calculated with discretized app.
Base+CSM: 30 Gaussian basis and =20 deg.
Continuum Level Density of 3α systemContinuum Level Density of 3α system
022
033
022
033
'
1111TrIm
1
)()()()()(
BBBB
BBBB
HEHEHEHE
EEEEE
)()( 1)point(
81
3
12
ClBe
OCMClNG
iiB VVTtH
)()( 1)point(
81)point(
3
1
02
ClBe
ClG
iiB VVTtH
α 1
α 2
α 3
1
1
8Be• α1- α2: resonance + continuum
• (α1α2)- α3: continuum
• α1- α2: continuum
• (α1α2)- α3: continuum
[Ref.] S.Shlomo, NPA 539 (1992) 17.
(2+)
Continuum Level Continuum Level
02+
03+
04+
05+
4. Complex scaled Lippmann-Schwinger equation
EVH )( 0
H0=T+VC V; Short Range Interaction
000 EH
Solutions of Lippmann-Schwinger Equation
00
1
V
HE
( Ψ0; regular at origin)
Outgoing waves
Complex Scaling
00)(
)(
1
V
HE
A. Kruppa, R. Suzuki and K. Kato, phys. Rev.C75 (2007), 044602
●Lines : Runge-Kutta method●Circles : CSM+Base
4He: (3He+p)+(3He+n) Coupled-Channel Model
Complex-scaled Lippmann-Schwinger Eq.
Direct breakup
Final state interaction (FSI)
• CSLM solution
• B(E1) Strength
Two-neutron distribution of 6He
(T. Aumann et.al, PRC59(1999)1252)
(T. Aumann et.al, PRC59(1999)1252)
Summary and conclusion
• The resolution of identity in the complex scaling method is presented to treat the resonant states in the same way as bound states.
• The complex scaling method is shown to describe not only resonant states but also continuum states on the rotated branch cuts.
• We presented several applications of the extended resolution of identity in the complex scaling method; strength functions of the Coulomb break reactions, continuum level density and three-body scattering states.
• Many-body resonant states of He-isotopes are studied.
Collaborators
Y. Kikuchi, K. Yamamoto, A. Wano, T. Myo, M. Takashina, C. Kurokawa, R. Suzuki, K. Arai, H. Masui, S. Aoyama, K. Ikeda, A. Kruppa. B. Giraud
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