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Single channel systems Coupled-channel systems (2) Three-body problems; Faddeev A=C1+C2+C3 Decay channels of A

（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 severaldefinitions of resonances

“Quantum Mechanics” by L.I. Schiff

… If any one of klis such that the denominator ( f(kl) ) of the expression for tanl,

|tanl| = | g(kl)/f(kl) | ∞ ,

( Sl(k) = e2il(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

“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.

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．

The solution φl(r) of the Schrödinger equation;

Satisfying the boundary conditions

,

the solution φl(r) is written as

where Jost solutions f±(k, r) is difined as

and Jost functions f±(k)

Then the S-matrix is expressed as

The important properties of the Jost functions:

1.

2.

From these properties, we have unitarity of the S-matrix;

- Ref.
- J. Humblet and L. Rosenfeld, Nucl. Phys. 26 (1961), 529-578
- L. Rosenfeld, Nucl. Phys. 26 (1961), 594-607.
- J. Humblet, Nucl. Phys. 31 (1962), 544-549.
- J. Humblet, Nucl. Phys. 50 (1964), 1-16.
- J. Humblet, Nucl. Phys. 57 (1964), 386-401.
- J.P. Jeukenne, Nucl. Phys. 58 (1964), 1-9
- J. Humblet, Nucl. Phys. A151 (1970), 225-242.
- J. Humblet, Nucl. Phys. A187 (1972), 65-95.

（2） Many-body resonance states

- （１）Two-body problems; easily solved

A[C1-C2]B+C3, Eth(C3)

[C2-C3]B+C1, Eth(C1)

[C3-C1]B+C2, Eth(C2)

B [C1-C2]R+C3, Eth(C12)

[C2-C3]R+C1, Eth(C23)

[C3-C1]R+C2, Eth(C31)

C C1+C2+C3, Eth(3)

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 rei q

k ke-i q

Yq= U(q) Y(r)

=ei3/2 q Y(rei q)

H(q)= U(q) H U(q)-1

H

physical picture of the complex scaling method

Resonance state

The resonance wave function behaves asymptotically as

When the resonance energy is expressed as

This asymptoticdivergence of the resonance wave function causes difficulties in the resonance calculations.

In the method of complex scaling, a radialcoordinate r is transformed as

Then the asymptotic form of the resonance wave function becomes

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
- Complex Scaled Schoedinger Equation

- 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) cq is an L2-class function:
- ii) Eq is independent on q ( )