1. What is a resonance? K. Kato Hokkaido University Oct. 6, 2010 KEK Lecture (1)

2. （ 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

3. (i) Resonance cross section  (E) ～ ————— １ (E – E r ) 2 + Γ 2 /4 Breit-Wigner formula

4. “Quantum Mechanics” by L.I. Schiff (ii) Phase shift … If any one of k l is such that the denominator ( f(k l ) ) of the expression for tan  l, |tan  l | = | g(k l )/f(k l ) | ∞, ( S l (k) = e 2i  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

5. Phase shift of 16 O + α OCM

6. “Theoretical Nuclear Physics” by J.M. Blatt and V.F. Weisskopf (iii) Decaying state We obtain a quasi-stational state if we postulate that for r>R c the solution consists of outgoing waves only. This is equivalent to the condition B=0 in ψ (r) = A e ikr + B e -ikr (for r >R c ). This restriction again singles out certain define solutions which describe the “decaying states” and their eigenvalues.

7. Resonance wave function For the resonance momentum k r =κ–iγ, ψ(r) = e i κr e rγ, (not normalizable (γ>0) ）

8. 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=E r – 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 ．

9. 4. Poles of S-matrix The solution φ l (r) of the Schrödinger equation; Satisfying the boundary conditions, the solution φ l (r) is written as

10. 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;

11. The pole distribution of the S-matrix in the momentum plane

12. The Riemann surface for the complex energy: E=k 2 /2

13. 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.

14. （ 2 ） Many-body resonance states （１） Two-body problems; easily solved Single channel systems Coupled-channel systems (2) Three-body problems; Faddeev A=C 1 +C 2 +C 3 Decay channels of A

15. A [C 1 -C 2 ] B +C 3, E th (C 3 ) [C 2 -C 3 ] B +C 1, E th (C 1 ) [C 3 -C 1 ] B +C 2, E th (C 2 ) B [C 1 -C 2 ] R +C 3, E th (C 12 ) [C 2 -C 3 ] R +C 1, E th (C 23 ) [C 3 -C 1 ] R +C 2, E th (C 31 ) C C 1 +C 2 +C 3, E th (3)

16. E th (C 3 ） E th (C 2 ) E th (C 2 ) E th (3) E th (C 32 ） E th (C 23 ） Multi-Riemann sheet E th (C 31 ） (3) N-Body problem; more complex 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。

17. Eigenvalues of H(  in the complex energy plane Complex scaling U(  r re i   k ke -i     U(   (r) =e i3/2   ( re i  ) H(  )= U(   U(    H

19. Complex Scaling Method physical picture of the complex scaling method Resonance state The resonance wave function behaves asymptotically as When the resonance energy is expressed as

20. the corresponding momentum is and the asymptotic resonance wave function Diverge!

21. 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 Then the asymptotic form of the resonance wave function becomes Converge!

22. It is now apparent that when π/2>(θ-θ r )>0 the wave function converges asymptotically. This result leads to the conclusion that the resonance parameters (E r, 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.

23. Eigenvalue Problem of the Complex Scaled Hamiltonian Complex scaling transformation Complex Scaled Schoedinger Equation

24. 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)   is an L 2 -class function: ii) E  is independent on 