1. Structure of Resonance and Continuum States Hokkaido University Unbound Nuclei Workshop Pisa, Nov. 3-5, 2008

2. 1. Resolution of Identity in Complex Scaling Method Continuum st. Spectrum of Hamiltonian Bound st. Resonant st. R R.G. Newton, J. Math. Phys. 1 (1960), 319 Completeness Relation (Resolution of Identity) Non-Resonant st.

3. Among the continuum states, resonant states are considered as an extension of bound states because they result from correlations and interactions. From this point of view, Berggren said In the present paper,* ) we investigate the properties** ) of resonant states and find them in many ways quite analogous to those of the ordinary bound states. *) NPA 109 (1968), 265. **) orthogonality and completeness

4. Separation of resonant states from continuum states Deformation of the contour Resonant states Ya.B. Zeldovich, Sov. Phys. JETP 12, 542 (1961). N. Hokkyo, Prog. Theor. Phys. 33, 1116 (1965). Convergence Factor Method Matrix elements of resonant states T. Berggren, Nucl. Phys. A 109, 265 (1968) Deformed continuum states

5. Complex scaling method coordinate: momentum: r B. Gyarmati and T. Vertse, Nucl. Phys. A160, 523 (1971). re iθ T. Myo, A. Ohnishi and K. Kato. Prog. Theor. Phys. 99(1998)801] inclination of the semi-circle

6. k k EE Single Channel system b1b1 b2b2 b3b3 r1r1 r2r2 r3r3 Coupled Channel systemThree-body system E| B.Giraud, K.Kato and A. Ohnishi, J. of Phys. A37 (2004),11575 B.Giraud and K.Kato, Ann.of Phys. 308 (2003), 115. Resolution of Identity in Complex Scaling Method

7. (Complex scaled) Structures of three-body continuum states

8. 0+0+ 1 Physical Importance of Resonant States M. Homma, T. Myo and K. Kato, Prog. Theor. Phys. 97 (1997), 561. red: 0 + blue: 1 -

9. B.S. R.S. Contributions from B.S. and R.S. to the Sum rule value S exc =1.5 e 2 fm 2 MeV Kiyoshi Kato

10. (A) Cluster Orbital Shell Model (COSM) Y. Suzuki and K. Ikeda, Phys. Rev. C38 (1988), 410 Core+Xn system The total Hamiltonian: where H C : the Hamiltonian of the core cluster A C U i : the interaction between the core and the valence neutron (Folding pot.) v ij : the interaction between the valence neutrons (Minnesota force, Av8, …) X Complex Scaled COSM

11. which has a peak in a region : The two-neutron distance : (B) Extended Cluster Model T-type coordinate system The di-neutron like correlation between valence neutrons moving in the spatially wide region Y. Tosaka, Y Suzuki and K. Ikeda; Prog. Theor. Phys. 83 (1990), 1140. K. Ikeda; Nucl. Phys. A538 (1992), 355c. When R~ 5-7 fm, to describe the short range correlation accurately up to 0.5 fm, the maximum -value is 10~14. θ

12. (C) Hybrid-TV Model S. Aoyama, S. Mukai, K. Kato and K. Ikeda, Prog. Theor. Phys. 94, 343-352 (1995) + (p 1/2 ) 2 (p 3/2 ) 2 Rapid convergence!! (p,sd)+T-base

13. Two-neutron density distribution of 6 He Hybrid-TV model (COSM 9ch + ECM 1ch) Harmonic oscillator (0p 3/2 only) S=0 S=1 Total (0p 3/2 ) Hybrid-TV

14. H.Masui, K. Kato and K.Ikeda, PRC75 (2007), 034316. 18 O 6 He

15. Excitation of two-neutron halo nuclei (Borromean nuclei) Soft-dipole mode Structure of three-body continuum Three-body resonant states Complex scaling method Resonant state Bound state (divergent) (no-divergent) S. Aoyama, T. Myo, K, Kato and K. Ikeda; Prog. Theor. Phys. 116, (2006) 1.

17. Y. Aoyama Phys. Rev. C68 (2003) 034313. 1 - ( Soft Dipole Resonance) pole in 4 He+n+n (CSM+ACCC) 1 - resonant state?? It is difficult to observe as an isolated resonant state!! E r ~3 MeV Γ~32 MeV

19. 7 He: 4 He+n+n+n COSM T. Myo, K. Kato and K. Ikeda, PRC76 (2007), 054309

20. Coulomb breakup reaction 3. Coulomb breakup reactions of Borromean systems Structures of three-body continuum state

21. Strength Functions of Coulomb Breakup Reaction

22. 9 Li+n+n 10 Li(1 + )+n 10 Li(2 + )+n Resonances T. Myo, A. Ohnishi and K. Kato, Prog. Theor. Phys. 99 (1998), 801. in CSM

23. T. Myo, K. Kato, S. Aoyama and K. Ikeda, PRC63(2001), 054313

25. coupled channel [ 9 Li+n+n] + [ 9 Li * +n+n] PRL 96, 252502 (2006) T. Myo

26. A.T.Kruppa, Phys. Lett. B 431 (1998), 237-241 A.T. Kruppa and K. Arai, Phys. Rev. A59 (1999), 2556 K. Arai and A.T. Kruppa, Phys. Rev. C 60 (1999) 064315 Definition of LD: 4. Unified Description of Bound and Unbound States Continuum Level Density

27. 1 Resonance: Rotated Continuum: Descretization RI in complex scaling

28. 2θ EE εIεI εIεI

29. Continuum Level Density: Basis function method:

30. Phase shift calculation in the complex scaled basis function method In a single channel case, S.Shlomo, Nucl. Phys. A539 (1992), 17.

31. Phase shift of 8 Be= + calculated with discretized app. Base+CSM: 30 Gaussian basis and =20 deg.

32. Description of unbound states in the Complex Scaling Method H 0 =T+V C V Short Range Interaction Solutions of Lippmann-Schwinger Equation Ψ 0 ; regular at origin Outgoing waves Complex Scaling A. Kruppa, R. Suzuki and K. Kato, phys. Rev.C75 (2007), 044602

33. T-matrix T l (k) Second term is approximated as where T l (k)

34. Lines : Runge-Kutta method Circles : CSM+Base

35. Complex-scaled Lippmann-Schwinger Eq. Direct breakup Final state interaction (FSI) CSLM solution B(E1) Strength

36. Dalitz distribution of 6 He Decay process –Di-neutron-like decay is not seen clearly.

37. . Summary and conclusion It is shown that resonant states play an important role in the continuum phenomena. The resolution of identity in the complex scaling method is presented to treat the three-body resonant states in the same way as bound states. The complex scaling method is shown to describe not only resonant states but also non-resonant continuum states on the rotated branch cuts. We presented several applications of the extended resolution of identity in the complex scaling method; sum rule, break-up strength function and continuum level density.

38. Collaboration: S. Aoyama(Niigata Univ.), H. Masui(Kitami I. T.), T. Myo (Osaka Tech. Univ.), R. Suzuki(Hokkaido Univ.), C. Kurokawa(Juntendo Univ.), K. Ikeda(RIKEN) Y. Kikuchi(Hokkaido Univ.)