Jost関数法と共鳴部分幅および仮想状態

(1) Jost 関数法 (Jost Function Method)(2)共鳴部分幅 （Partial Widths)(3)仮想状態 (Virtual States)(3)仮想状態 (Virtual States)

Table I. Values of the resonant poles of the Noro-Taylor model.

pole Er (a.u.) Γ (a.u.)

1 4.768197 1.420192 ×10 -3

2 7.241200 1.511912

3 8.171216 6.508332

4 8.440526 12.56299

5 8.072642 19.14563

6 7.123813 26.02534

7 5.641023 33.07014

8 3 662702 40 194678 3.662702 40.19467

9 1.220763 47.33935

10 1 658115 54 4608710 -1.658115 54.46087

11 -4.950418 61.52509

12 -8.635939 68.50621

Partial Decay Widths

Channel radius dependence

Definition of partial widthsDefinition of partial widths

)()(lim )( rkHAr cccresr

+

∞→ ∑=Ψcr ∞→

2AkμΓ22212 cccc

AA

kkμ

=Γ

Γ∑ Γ=Γ N

n1121 cccc AkμΓ =n n1

N. Moiseyv and U. Peskin; Phys. Rev. A42(1990) 255.

Partial widths of resonant statesPartial widths of resonant states

Jost Function Method;S A Sofianos and S A RakityanskyS.A. Sofianos and S.A. Rakityansky

J. Phys. A: Math. Gen. 30(1997), 3725,J Ph A M th G 31(1998) 5149J. Phys. A: Math. Gen. 31(1998), 5149.

}{)( )()()()()()( −−++±∂Nn FHFHVHEF ∓μ }{),( )(

')(

')(

')(

'1' ')(

2)( −−++

=

± +±=∂ ∑ mnnmnnN

n nnnn

nnmr FHFHVH

kirEF ∓μ

: Homogeneous solutions ),()( rkH nn±

0),(det )( =∞− EF : Resonances

Partial Width ΓΓPartial Width2/iEE

i)E(S)E(Sr

'nn'nn

Bn'nn Γ+−

ΓΓ−δ=

|||)(lim| nnnnresn SSEE =−=Γ |||)(

lim|''''' nnnnres

resEEn SSEE

=−

=Γ →

[ ] ∑+

+ resE

EGEFF )()()()(

)([ ][ ] ∑

+

∑

+

−

== m resmnresnm

resEnn

EGEFEGEF

FF

)()()()(

)()(

)(

[ ] ∑− m resmnresmn

nn

EGEFF

F )()( ''''

)(

)}(d t{)(),( )(

1)( =∞−

−−

EFEGEF nm

nm )},(det{)( )( ∞− EFnm

Current density method for partial widthsN. Moiseyev and U. Peskin; Phys. Rev. A42(1990) 255.

Current density method for partial widths

)(r∑= ϕψ

N. Moiseyev and U. Peskin; Phys. Rev. A42(1990) 255.

),(1

rn nres∑=

= ϕψ

)()( jA→ ),()( rjAr nnrn ∞→ϕ

),()( )( rkHrj nkn

n+= μ ),()( kj nnkn

Partial Width:2

' nnnn

AA

kkμ

=ΓΓ

''' nnnn AkμΓ

T-matrix schemeT-matrix scheme

)f( )|V|(φΓ 2)f(

res)f(

nn |)|V|()|V|(|

'' ψφψφ

=ΓΓ

resnn)|V|( '' ψφΓ

∑ϕ=ψ nres )r()r(n

)rk(Hk)r( n)(

nn

n)f(n

+μ=φ

∫ ∑ ϕμ=ψφ −

'n'n'nnn

)(

n

nres

)f(n )r(V)rk(drHk)|V|(

∫ ∑ ∑ φμ= −m'nm'nnn

)(n )r(cV)rk(drHk ∫ ∑ ∑ φ'n

m'nm

m'nnnn

)r(cV)rk(drHk

∫⎤⎡μ ∑ ∫ ∑ ⎥⎦

⎤⎢⎣

⎡φμ= −

m 'nm'n'nnn

)(m

n

n )r(V)rk(drHck

),E(Fcki res)(

nmm

mn

n ∞μ= +∑m

)r(a)r( )f(φ→ϕ )r(a)r( )(nnrn φ⎯⎯→⎯ϕ ∞→

)r(cli m

nmm⎥⎤

⎢⎡ φ∑

)r(lima )f(

n

m

rn

⎥⎥⎥

⎦⎢⎢⎢

⎣φ

=∞→

)}r,E(F)rk(H)r,E(F)rk(H{c1r

)(nmn

)(nr

)(nmn

)(nm ⎥

⎤⎢⎡ +

⎥⎦⎢⎣

−−++∑

)k(H

)},()(),()({c2lim

)(n

mrnmnnrnmnnm

r ⎥⎥⎥

⎢⎢⎢

μ=

+∞→

∑

1

)rk(Hk n)(

nn

n⎥⎥⎦⎢

⎢⎣

μ +

),E(Fck21

r)(

nmm

mn

n ∞μ= +∑m

)rk(HAlim)r(lim n)(

nnrresr ∑ +

∞→∞→=ψ

)r,E(Fc)rk(Hlim res)(

nmmn)(

n

n

∑ ∑ ++= )()(n m

resnmmnnr ∑ ∑∞→

∴ ),E(FcNA res)(

nmmn ∞= ∑ + ),( resm

nmmn ∑

ΓΓ N∑=Γ=Γ

n n1

Jost Function MethodJost Function Method

+ Complex Scaling Method+ Complex Scaling Method

Complex Scaled Jost Function Method;Complex Scaled Jost Function Method;(CSJFM)

Application to a three body resonance

5He: 4He+n

H M i S A T M K K t d K Ik d N l Ph A673 (2000) 207H. Masui, S. Aoyama, T. Myo, K. Kato and K. Ikeda, Nucl. Phys. A673 (2000), 207

10Li: 9Li+n