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Unified Description of Bound and Unbound States -- Resolution of Identity --

Unified Description of Bound and Unbound States -- Resolution of Identity --. KEK Lecture (2). K. Kato Hokkaido University Oct. 6, 2010. 1. Resolution of Identity in the Complex Scaling Method. Completeness Relation (Resolution of Identity). R.G. Newton, J. Math. Phys. 1 (1960), 319.

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Unified Description of Bound and Unbound States -- Resolution of Identity --

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

  2. 1. Resolution of Identity in the Complex Scaling Method Completeness Relation (Resolution of Identity) R.G. Newton, J. Math. Phys. 1 (1960), 319 Bound states Continuum states Ｌ Resonant states non-resonant continuum states 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 Deformed continuum states Resonant states T. Berggren, Nucl. Phys. A 109, 265 (1968) Deformation of the contour Matrix elements of resonant states Convergence Factor Method Ya.B. Zel’dovich, Sov. Phys. JETP 12, 542 (1961). N. Hokkyo, Prog. Theor. Phys. 33, 1116 (1965).

  4. J. Aguilar and J.M.Combes, J. Math. Phys. 22, 269 (1971) E.Balslev and J.M.Combes, J. Math. Phys. 22, 280 (1971) Complex scaling method reiθ coordinate: r B. Gyarmati and T. Vertse, Nucl. Phys. A160, 523 (1971). momentum: Resonant states Rotated Continuum states T. Myo, A. Ohnishi and K. Kato. Prog. Theor. Phys. 99(1998)801]

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

  6. Eigenvalues of H(θ) with a L2 basis set (L2 basis set ; Gaussian basis functions)

  7. E| E| b3 b2 b1 r2 r3 r1 B.Giraud, K.Kato and A. Ohnishi, J. of Phys. A37 (2004),11575 Coupled Channel system Three-body system Bany-body system

  8. 10Li :9Li(3/2-) +n(p3/2) T. Myo, A. Ohnishi and K. Kato, Prog. Theor. Phys. 99 (1998), 801. in CSM 10Li(1+)+n 10Li(2+)+n Resonances 9Li+n+n

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

  10. 2. Strength Functions and Coulomb Breakup Reaction

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

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

  13. Coulomb breakup strength of 6He 6He : 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 11Li T. Nakamura et al., Phys. Rev. Lett. 96, 252502 (2006)

  15. 3. Continuum level density Definition of LD: 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

  16. 1 RI in complexscaling Resonance: Continuum: Discreet distribution

  17. Discretized Continuum States in the Complex Scaling Method εI εI E E 2θ 2θ

  18. Continuum Level Density: Basis function method:

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

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

  21. Continuum Level Density of 3α system [Ref.] S.Shlomo, NPA 539 (1992) 17. 8Be α２ • α1- α2: resonance + continuum • (α1α2)- α3: continuum α１ α３ • α1- α2: continuum • (α1α2)- α3: continuum

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

  23. 4. Complex scaled Lippmann-Schwinger equation H0=T+VC V；　Short Range Interaction （Ψ0; regular at origin） Solutions of Lippmann-Schwinger Equation Complex Scaling Outgoing waves A. Kruppa, R. Suzuki and K. Kato, phys. Rev.C75 (2007), 044602

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

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

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

  27. (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

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