Transcript ６．Jost関数法と共鳴部分幅および仮想状態

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)
Table I. Values of the resonant poles of the Noro-Taylor model.
pole
Er (a.u.)
Γ (a.u.)
1
4.768197
2
7.241200
1.511912
3
8.171216
6.508332
4
8.440526
12.56299
5
8.072642
19.14563
6
7.123813
7
5.641023
33.07014
8
3.662702
40.19467
9
1.220763
47.33935
10
-1.658115
54.46087
11
-4.950418
61.52509
12
-8.635939
68.50621
1.420192 ×10 -3
26.02534
Partial Decay Widths
Channel radius dependence
Definition of partial widths
lim res (r )   Ac H
r 
()
c
( kc r )
c
 c kc A

c
 c kc A
c2
1
2
1
2
1
2
c2
2
c1

N

n 1
n
N. Moiseyv and U. Peskin; Phys. Rev. A42(1990) 255.
Partial widths of resonant states
Jost Function Method;
S.A. Sofianos and S.A. Rakityansky
J. Phys. A: Math. Gen. 30(1997), 3725,
J. Phys. A: Math. Gen. 31(1998), 5149.
()
 r Fnm
( E, r )

n
i 2 k n
() N
H n  Vnn'{H n( ' ) Fn('m)
n '1
H n(  ) (kn , r ) : Homogeneous solutions
det F
( )
( E , )  0 : Resonances
 H n( ' ) Fn('m) }
Partial Width
Snn ' (E)  S (E) nn '
B
n
n n '
i
E  E r  i / 2
n
( E  Eres ) S nn
S nn
| Elim
|

|
|
 Eres
n '
( E  Eres ) S n 'n ' S n 'n '
F


F
()


Eres
F
()
nn
Eres
()
F
()

m
F ( Eres )Gmn ( Eres )
()
n 'm
F ( Eres )Gmn' ( Eres )
n 'n '
1
F ( E , ) nm
()


m
()
nm
Gnm ( E )

()
det{F ( E , )}
Current density method for partial widths
N. Moiseyev and U. Peskin; Phys. Rev. A42(1990) 255.
 res  n1 n (r ),
 n (r ) 
A
j
(
r
),
r 
n n
jn (r ) 
n
()
H
(
k
,
r
)
n
kn
Partial Width: n   n ' k n An
n '
 n k n ' An '
2
T-matrix scheme
n ( | V |  res ) 2
|
|
n ' ( | V |  res )
(f )
n
(f )
n'
 res (r)   n (r)
n

n
 (r ) 
(f )
n
k n
H (n ) (k n r )
((nf ) | V |  res ) 


i
n
n
n
( )
drH
(k n r ) Vnn 'n ' (r )

k n
k n
n'
( )
drH
(k n r ) Vnn '  c m n 'm (r )

n'
m


( )
c m  drH (k n r ) Vnn 'n 'm (r)
k n 
m
n'


n
k n
c
m
m
()
Fnm
(E res , )
n (r) r
a  (r)

(f )
n n
  c m nm (r ) 

a n  lim m ( f )
r  
 n (r ) 


1

()
()
()
()
 2  c m {H n (k n r )Fnm (E r , r )  H n (k n r )Fnm (E r , r )}
 lim m

r 
n


H (n ) (k n r )
k n


1 k n
()

c
F
( E r , )

m
nm
n m
2
lim  res (r )  lim  A n H (k n r )
r 
r 
()
n
n
 lim  H (k n r ) c F (E res , r )
r 

()
n
n
()
m nm
m
An  N c F (E res , )
()
m nm
m

N

n 1
n
Jost Function Method
+ Complex Scaling Method
Complex Scaled Jost Function Method;
(CSJFM)
Application to a three body resonance
5He: 4He+n
H. Masui, S. Aoyama, T. Myo, K. Kato and K. Ikeda, Nucl. Phys. A673 (2000), 207
10Li: 9Li+n