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