1. Unified Description of Bound and Unbound States -- Resolution of Identity -- K. Kato Hokkaido University Oct. 6, 2010 KEK Lecture (2)

2. 1. Resolution of Identity in the Complex Scaling Method Continuum statesBound states Resonant states non-resonant continuum states Ｌ R.G. Newton, J. Math. Phys. 1 (1960), 319 Completeness Relation (Resolution of Identity) Among the continuum states, resonant states are considered as an extension of bound states because they result from correlations and interactions.

3. Separation of resonant states from continuum states Deformation of the contour Resonant states Ya.B. Zel’dovich, 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

4. 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] Rotated Continuum statesResonant states J. Aguilar and J.M.Combes, J. Math. Phys. 22, 269 (1971) E.Balslev and J.M.Combes, J. Math. Phys. 22, 280 (1971)

5. k k EE Single Channel system B.Giraud and K.Kato, Ann.of Phys. 308 (2003), 115. Resolution of Identity in Complex Scaling Method

6. Eigenvalues of H(θ) with a L 2 basis set (L 2 basis set ; Gaussian basis functions)

7. b1b1 b2b2 b3b3 r1r1 r2r2 r3r3 Coupled Channel system Three-body system E| B.Giraud, K.Kato and A. Ohnishi, J. of Phys. A37 (2004),11575 Bany-body system

8. 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 10 Li : 9 Li(3/2 - ) +n(p 3/2 )

9. Complex Scaled Green’s Functions Green’s operator Resolution of Identity Complex Scaled Green’s function Complex scaled Green’s operator

10. 2. Strength Functions and Coulomb Breakup Reaction

11. Coulomb Breakup Reactions of Three-Body Systems (two- neutron halo systems) Breakup Mechanism: Direct or Sequential ? Simultaneous description of Structure and Reaction Neutron-correlation in 2-neutron halo states

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

13. Coulomb breakup strength of 6 He 6 He : 240MeV/A, Pb Target (T. Aumann et.al, PRC59(1999)1252) T.Myo, K. Kato, S. Aoyama and K. Ikeda PRC63(2001)054313.

14. Coulomb breakup cross section of 11 Li T. Nakamura et al., Phys. Rev. Lett. 96, 252502 (2006)

15. 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: 3. Continuum level density

16. 1 Resonance: Continuum: Discreet distribution RI in complex scaling

17. 2θ EE εIεI εIεI Discretized Continuum States in the Complex Scaling Method

18. Continuum Level Density: Basis function method:

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

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

21. Continuum Level Density of 3α system α１α１ α２α２ α３α３ 8 Be α 1 - α 2 : resonance + continuum (α 1 α 2 )- α 3 : continuum α 1 - α 2 : continuum (α 1 α 2 )- α 3 : continuum [Ref.] S.Shlomo, NPA 539 (1992) 17.

22. (2 +) Continuum Level 02+02+ 03+03+ 04+04+ 05+05+

23. 4. Complex scaled Lippmann-Schwinger equation 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

24. ● Lines : Runge-Kutta method ● Circles : CSM+Base 4 He: ( 3 He+p)+( 3 He+n) Coupled-Channel Model

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

26. Two-neutron distribution of 6 He (T. Aumann et.al, PRC59(1999)1252)

28. Summary and conclusion The resolution of identity in the complex scaling method is presented to treat the resonant states in the same way as bound states. The complex scaling method is shown to describe not only resonant states but also continuum states on the rotated branch cuts. We presented several applications of the extended resolution of identity in the complex scaling method; strength functions of the Coulomb break reactions, continuum level density and three-body scattering states. Many-body resonant states of He-isotopes are studied.

29. Collaborators Y. Kikuchi, K. Yamamoto, A. Wano, T. Myo, M. Takashina, C. Kurokawa, R. Suzuki, K. Arai, H. Masui, S. Aoyama, K. Ikeda, A. Kruppa. B. Giraud