Structure of Resonance and

Continuum States

Hokkaido University

Unbound Nuclei Workshop

Pisa, Nov. 3-5, 2008

1. Resolution of Identity in Complex Scaling Method

Continuum st.

Spectrum of

Hamiltonian

Bound st.

Resonant st.

|~|~|1 1kkRnn

bn

dkuu

rr

e

rr

e

RR.G. Newton, J. Math. Phys. 1 (1960), 319

Completeness Relation (Resolution of Identity)

Non-Resonant st.

Among the continuum states, resonant states are considered as an extension of bound states because they result from correlations and interactions.

From this point of view, Berggren said

“In the present paper,*) we investigate the properties**) of resonant states and find them in many ways quite analogous to those of the ordinary bound states.”

*) NPA 109 (1968), 265. 　 **) orthogonality and completeness

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

kkLnn

N

rnnn

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]

|~|~|~|1 1

kkL

N

rnnnnn

bnk

r

dkuuuu

inclination of the semi-circle

k k E E

Single Channel system

b1b2b3 r1r2 r3

Coupled Channel system Three-body system

E| E|

B.Giraud, K.Kato and A. Ohnishi, J. of Phys. A37 (2004),11575

B.Giraud and K.Kato, Ann.of Phys. 308 (2003), 115.

Resolution of Identity in Complex Scaling Method

0 0

0

(Complex scaled)

Structures of three-body continuum states

　　　　　　　　　　　　　　　

　　　　　　　　　　　　　　　　　　

　　　　

　　　　

　　　　

0+1 －

Physical Importance

of Resonant States

M. Homma, T. Myo and K. Kato, Prog. Theor. Phys. 97 (1997), 561.

red: 0+

blue: 1-

B.S.

R.S.

Contributions from B.S. and R.S. to the Sum rule value

Sexc=1.5e2fm2MeV

• Kiyoshi Kato

(A) Cluster Orbital Shell Model (COSM)

• Y. Suzuki and K. Ikeda, Phys. Rev. C38 (1988), 410

Core+Xn system

The total Hamiltonian:

X

jiji

cij

X

iii

X

iC mA

vUm

HH ppp)1(

1

2

1

1

2

1

where

HC : the Hamiltonian of the core cluster AC

Ui : the interaction between the core and

the valence neutron (Folding pot.)

vij : the interaction between the valence

neutrons 　 (Minnesota force, Av8, …)

1

2

i

X X

２． Complex Scaled COSM

which has a peak in a region :

The two-neutron distance :

(B) Extended Cluster Model ー T-type coordinate system ー

),(cos)][:2( 20 Pn Ji

1

The di-neutron like correlation between valence neutrons moving in the spatially wide region

Y. Tosaka, Y Suzuki and K. Ikeda; Prog. Theor. Phys. 83 (1990), 1140.K. Ikeda; Nucl. Phys. A538 (1992), 355c.

When R~5-7fm, to describe the short range correlation accurately up to 0.5 fm, the maximum -value is 10~14.

1

Rd

θ

(C) Hybrid-TV Model

S. Aoyama, S. Mukai, K. Kato and K. Ikeda, Prog. Theor. Phys. 94, 343-352 (1995)

+

(p1/2)2(p3/2)2

2)14(

Rapid convergence!!(p,sd)+T-base

Two-neutron density distribution of 6HeH

ybrid

-TV

model (C

OSM

9ch

+ E

CM

1ch

)H

arm

onic

osc

illato

r (0

p3/2 o

nly

)

S=0

S=1

Total

(0p3/2) ２ Hybrid-TV

H.Masui, K. Kato and K.Ikeda, PRC75 (2007), 034316.

18O

6He

Excitation of two-neutron halo nuclei (Borromean nuclei)

Soft-dipole mode

Structure of three-body continuum

Three-body resonant states

Complex scaling method

Resonant state 　 Bound state

(divergent) (no-divergent)

S. Aoyama, T. Myo, K, Kato and K. Ikeda; Prog. Theor. Phys. 116, (2006) 1.

Y. Aoyama ； Phys. Rev. C68 (2003) 034313.

1- ( Soft Dipole Resonance) pole in 4He+n+n (CSM+ACCC)

1- resonant state??It is difficult to observe as an isolated resonant state!!

Er~3 MeV

Γ~32 MeV

)()()1( rVrV

7He: 4He+n+n+n COSM

T. Myo, K. Kato and K. Ikeda, PRC76 (2007), 054309

Coulomb breakup reaction

3. Coulomb breakup reactions of Borromean systems

Structures of three-body continuum state

　　　　　　　　　

　　　　　　

Strength Functions of Coulomb Breakup Reaction

|~|"|~|'|~|~|~|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

T. Myo, K. Kato, S. Aoyama and K. Ikeda, PRC63(2001), 　 054313

coupled channel [9Li+n+n]

+

[9Li*+n+n]

PRL 96, 252502 (2006)

T. Myo

)()( 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

4. Unified Description of Bound and Unbound States

Continuum Level Density

B

B

B

R RB

N

n

N

nL C

CRn

Bn EE

dEEEEE

iHETrE

111Im

1

1Im

1)(

1

Resonance:

Rotated Continuum:2

R

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)(

Descretization

RI in complex

scaling

2θ

2θ

E E

εI εI

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.

Description of unbound states in the Complex Scaling Method

EVH )( 0

H0=T+VC V ；　 Short Range Interaction

000 EH

Solutions of Lippmann-Schwinger Equation

00

1

V

iHE

（ Ψ0; regular at origin ）

Outgoing waves

Complex Scaling

00)(

)(

1

V

HE

A. Kruppa, R. Suzuki and K. Kato, phys. Rev.C75 (2007), 044602

T-matrix

Tl(k) ＝

Second term is approximated as

where

Tl(k)

●Lines : Runge-Kutta method●Circles : CSM+Base

Complex-scaled Lippmann-Schwinger Eq.

Direct breakup

Final state interaction (FSI)

• CSLM solution

• B(E1) Strength

Dalitz distribution of 6He

• Decay process– Di-neutron-like decay is not seen clearly.

coskkmA

EEEc

22

2

21

６ . Summary and conclusion• It is shown that resonant states play an important role in

the continuum phenomena.

• The resolution of identity in the complex scaling method is presented to treat the three-body resonant states in the same way as bound states.

• The complex scaling method is shown to describe not only resonant states but also non-resonant continuum states on the rotated branch cuts.

• We presented several applications of the extended resolution of identity in the complex scaling method; sum rule, break-up strength function and continuum level density.

Collaboration:

　 S. Aoyama(Niigata Univ.), 　　　 H. Masui(Kitami I. T.),

　 T. Myo (Osaka Tech. Univ.), R. Suzuki(Hokkaido Univ.),

C. Kurokawa(Juntendo Univ.), K. Ikeda(RIKEN)

Y. Kikuchi(Hokkaido Univ.)