What is a resonance?
KEK Lecture (1)
K. Kato
Hokkaido University
Oct. 6, 2010
（1） What is a resonance？
The discrete energy state created in the
continuum energy region by the
interaction, which has an outgoing
boundary condition.
However, there are several definitions of
resonances
(i) Resonance cross section
s(E) ～
１
—————
(E – Er)2 + Γ2/4
Breit-Wigner formula
(ii) Phase shift
“Quantum Mechanics” by L.I. Schiff
… If any one of kl is such that the denominator
( f(kl) ) of the expression for tanl,
|tanl| = | g(kl)/f(kl) |
∞
,
( Sl(k) = e2il(k) ),
is very small, the l-th partial wave is said to be in
resonance with the scattering potential.
Then, the resonance:
l(k) = π/2 + n π
•Phase shift of 16O + α OCM
(iii) Decaying state
“Theoretical Nuclear Physics”
by J.M. Blatt and V.F. Weisskopf
We obtain a quasi-stational state if we postulate that
for r>Rc the solution consists of outgoing waves
only. This is equivalent to the condition B=0 in
ψ (r) = A eikr + B e-ikr
(for r >Rc).
This restriction again singles out certain define
solutions which describe the “decaying states” and
their eigenvalues.
•Resonance wave function
For the resonance momentum kr=κ–iγ,
ψ(r) = ei κr erγ,
(not normalizable (γ>0)）
G. Gamow, Constitution of atomic nuclei and dioactivity
(Oxford U.P., 1931)
A.F.J. Siegert, Phys. Rev. 56 (1939), 750.
The physical meaning of a complex energy
E=Er – iΓ/2
can be understood from the time depen-dence of the wave
function
ψ(t) = ψ(t=0) exp(－iEt/ｈ)
and its probability density
| ψ(t)|2 = |ψ(t=0)|2 exp(－Γt/２ｈ).
The lifetime of the resonant state is given by τ = ｈ/Γ．
4. Poles of S-matrix
The solution φl(r) of the Schrödinger equation;
d 2 l
2
l (l  1)
2
 { 2 V (r ) 
} l  k  l
2
2
dr

r
Satisfying the boundary conditions
lim r l 1l (k , r ) , 1
r 0
the solution φl(r) is written as
i
 l (k , r )  { f  (k ) f  (k , r )  f  (k ) f  (k , r )}
2k
if  (k ) ikr f  (k ) ikr

{e 
e }
r  2k
f  (k )
where Jost solutions f±(k, r) is difined as
 ikr
lim e
f  (k , r )  1,
r 
and Jost functions f±(k)
(2l  1) lim r l f  (k , r )  f  (k )
r 0
Then the S-matrix is expressed as
l f  (k )
S l (k )  (1)
.
f  (k )
The important properties of the Jost functions:
1. f  (k , r )  f  (k , r ),
f (k , r )  f  (k , r ),
2. f  (k )  f  (k ),
f (k )  f  (k ).
*

*

*
*
From these properties, we have unitarity of the S-matrix;
S *  k *  S  k   S  k  S  k   1.
The pole distribution of the S-matrix in the momentum plane
The Riemann surface for the complex energy:
E=k2/2
Ref.
1. J. Humblet and L. Rosenfeld, Nucl. Phys. 26 (1961),
529-578
2. L. Rosenfeld, Nucl. Phys. 26 (1961), 594-607.
3. J. Humblet, Nucl. Phys. 31 (1962), 544-549.
4. J. Humblet, Nucl. Phys. 50 (1964), 1-16.
5. J. Humblet, Nucl. Phys. 57 (1964), 386-401.
6. J.P. Jeukenne, Nucl. Phys. 58 (1964), 1-9
7. J. Humblet, Nucl. Phys. A151 (1970), 225-242.
8. J. Humblet, Nucl. Phys. A187 (1972), 65-95.
（2） Many-body resonance states
（１）Two-body problems; easily solved
Single channel systems
Coupled-channel systems
(2) Three-body problems; Faddeev
A=C1+C2+C3
Decay channels of A
A
B
C
[C1-C2]B+C3,
Eth(C3)
[C2-C3]B+C1,
Eth(C1)
[C3-C1]B+C2,
Eth(C2)
[C1-C2]R+C3,
Eth(C12)
[C2-C3]R+C1,
Eth(C23)
[C3-C1]R+C2,
Eth(C31)
C1+C2+C3,
Eth(3)
Multi-Riemann sheet
Eth(C3） Eth(C2) Eth(C2)
Eth(3)
Eth(C32）
Eth(C23）
Eth(C31）
様々な構造をもったクラスター閾値から始まる連続状態がエ
ネルギー軸上に縮退して観測される。
(3) N-Body problem; more complex
Eigenvalues of H(q in the complex energy plane
Complex scaling
U(q ; r
HY  EY
R
ikr
r ir
YH
r


e

e
e

r
 

(k    i )

ke-i q
Yq U(q Y(r)
=ei3/2 q Y(rei
q
)
H(q)= U(q H U(q1
H (q )Yq  EYq
iq
|k |sin(q q r ) r
ikre
YqR r


e

e

r
 0

k
rei q
 ei|k |cos(q q r ) r
(q r  tan -1  )

Complex Scaling Method
physical picture of the complex scaling method
Resonance state
The resonance wave function behaves asymptotically as
 (r )  e
r 
ikr
r  .
When the resonance energy is expressed as

E  Er  i ,
2

1 1  
q r  tan ( ) ,
2
2 Er 

the corresponding momentum is
k    i 
| k r | e
iq r
2E 
2 | E |e  iq r
,
and the asymptotic resonance wave function
 (r )  e
r 
i|k |exp( iq r ) r
e
i|k |r cosq r
|k |r sinq r
e
Diverge!
.
This asymptotic divergence of the resonance wave function
causes difficulties in the resonance calculations.
In the method of complex scaling, a radial coordinate r is
transformed as
U (q ); r  re ,
iq
 iq
p  pe .
Then the asymptotic form of the resonance wave function
becomes
 (r )  e
r 
i|k |e  iq r reiq
e
i|k |rei (q q r )
e
i|k |r cos(q q r )
e
|k |r sin(q q r )
Converge!
It is now apparent that when π/2>(θ-θr)>0 the
wave function converges asymptotically. This
result leads to the conclusion that the resonance
parameters (Er, Γ) can be obtained as an
eigenvalue of a bound-state type wave function.
This is an important reason why we use the
complex scaling method.
Eigenvalue Problem of the Complex Scaled
Hamiltonian
•
Complex scaling transformation
U(q)f (r )  e
•
i 3q / 2
f (re
iq
Complex Scaled Schoedinger Equation
H qq  E qq
1
H q  U(q)HU (q),
q  U(q)
H TV
)
ABC Theorem
J.Aguilar and J. M. Combes; Commun. Math. Phys. 22 (1971), 269.
E. Balslev and J.M. Combes; Commun. Math. Phys. 22(1971), 280.
i) q is an L2-class function:
 q   ci (q)u i ,
i
|| u i || 
ii) Eq is independent on q  q  1 arg( E res ) 
2
E
res
 E r  i / 2