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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