1. Application of correlated basis to a description of continuum states 19 th International IUPAP Conference on Few- Body Problems in Physics University of Bonn, Germany 31.08 – 05.09.2009 Wataru Horiuchi (Niigata, Japan) Yasuyuki Suzuki (Niigata, Japan)

2. Introduction Accurate solution with realistic interactions – Nuclear interaction – Nuclear structure – Some difficulties Realistic interaction (short-range repulsion, tensor) Continuum description → much more difficult (boundary conditions etc.) Contents – Our correlated basis – Method for describing continuum states from L 2 basis – Examples (n-p, alpha-n scattering) – Summary and future works

3. Variational calculation for many-body systems Hamiltonian Basis function Realistic nucleon-nucleon interactions: central, tensor, spin-orbit Generalized eigenvalue problem

4. Correlated Gaussian and global vector Global Vector Representation (GVR) x1x1 x3x3 x2x2 Correlated Gaussian Global vector Parity (-1) L 1 +L 2

5. Advantages of GVR No need to specify intermediate angular momenta. – Just specify total angular momentum L Nice property of coordinate transformation – Antisymmetrization, rearrangement channels Variational parameters A, u → Stochastically selected x1x1 x3x3 x2x2 y1y1 y2y2 y3y3

6. 4 He spectrum Good agreement with experiment without any model assumption 3 H+p, 3 He+n cluster structure appear W. H. and Y. Suzuki, PRC78, 034305(2008) P-waveS-wave 3 H+p 3 He+n Ground state energy Accuracy ～ 60 keV. H. Kamada et al., PRC64, 044001 (2001)

7. For describing continuum states Bound state approximation – Easy to handle (use of a square integrable (L 2 ) basis) – Good for a state with narrow width – Ill behavior of the asymptotics Continuum states – Can we construct them in the L 2 basis? Scattering phase shift

8. Formalism(1) Key quantity: Spectroscopic amplitude (SA) The wave function of the system with E A test wave function U(r): arbitrary local potential (cf. Coulomb) Inhomogeneous equation for y(r)

9. Formalism(2) The analytical solution G(r, r’): Green’s function SA solved with the Green’s function (SAGF) Phase shift: v(r): regular solution h(r): irregular solution

10. Test calculations Neutron-proton phase shift Minnesota potential (Central) Numerov SAGF Neutron-alpha phase shift Minnesota potential + spin-orbit Alpha particle → four-body cal. R-matrix SAGF The SAGF method reproduces phase shifts calculated with the other methods. Relative wave function

11. Improvement of the asymptotics Ill behaviors of the asymptotics are improved

12. α+n scattering with realistic interactions Interactions: AV8’ (Central, Tensor, Spin-orbit) Alpha particle → four-body cal. Single channel calculation with α+n 1/2 + → fair agreement 1/2 -, 3/2 - → fail to reproduce distorted configurations of alpha three-body force K. M. Nollett et al. PRL99, 022502 (2007) Green’s function Monte Carlo S. Quaglioni, P. Navratil, PRL101, 092501 (2008) NCSM/RGM

13. Summary and future works Global vector representation for few-body systems – A flexible basis (realistic interaction, cluster state) – Easy to transform a coordinate set SA solved with the Green’s function (SAGF) method – Easy (Just need SA) – Good accuracy Possible applications (in progress) – Coupled channel Alpha+n scattering with distorted configurations ( 4 He*+n, t+d, etc) – Extension of SAGF to three-body continuum states E1 response function (cf. 6 He in an alpha+n+n) – Complex scaling method (CSM) – Lorentz integral transform method (LIT) – Four-body continuum Four-body calculation with the GVR – Electroweak response functions in 4 He (LIT, CSM)

15. Decomposition of the phase shift V c : central, tensor, spin-orbit Neutron-alpha scattering with 1/2 +