R. Lazauskas Application of the complex-scaling method for few-body scattering 07/08/2019 R. Lazauskas

Introduction Bound states Scattering R. Lazauskas Configuration space wave functions extend to infinity. Wave function of finite size (bound to the box) Variational method, multiple ways to discretize wave function and solve the Schrödinger eq. Move to momentum space? singularities… 07/08/2019 R. Lazauskas

Complex scaling method (Resonances) J. Nuttal and H. L. Cohen, Phys. Rev. 188 (1969) 1542 Radial Schrödinger equation: Complex scaling: Im(k) Resonance j q Re(k) Exp. bound if q>f 07/08/2019 R. Lazauskas 3

Complex scaling method (Resonances) Extremely efficient accuracy for atomic problems (3-body problem) Y. K. Ho, Phys. Rev. A23 (1981) 2137. “Complex scaled” potential remains simple: 07/08/2019 4 R. Lazauskas 4

Complex scaling method (Resonances) Successful also for nuclear case: 12C resonances in (a+a+a) model R. Álvarez-Rodrıguez et al., Euro. Phys. J. A31 (2007) 303 P. Descouvemont J. Phys. G. 37 (2010) 6 6He and 6Li systems via (a+N+N) model N. Tanaka, Y. Suzuki and K. Varga, PRC 56 (1997) 562 Nonexistence of the “observable” multineutron resonances S.A. Sofianos, S.A. Rakityansky and G.P. Vermaak, J. Phys. G. 23 (1997) 1619 H. Witała and W. Glöckle, PRC 60 (1999) 024002 R. Lazauskas and J. Carbonell, Phys. Rev. C 72 (2005) 034003 R. Lazauskas and J. Carbonell, Phys. Rev. C 71, 044004 (2005) 07/08/2019 5 R. Lazauskas 5

Complex scaling method (Resonances) Analiticity of the potential, limitation of the scaling angle: Starts to diverge for q>p/(2n) 6 Hardcore propagates! 07/08/2019 R. Lazauskas 6

Complex scaling method (2b scattering) J. Nuttal and H. L. Cohen, Phys. Rev. 188 (1969) 1542 Driven radial Schrödinger equation: Complex scaling: Exp. bound by the short range pot. term Vs Exp. bound if q<p/2 Extraction of amplitudes: From the asymptote of the solution Using Green’s theorem 07/08/2019 R. Lazauskas 7

Complex scaling method (2b scattering) Ecm=1 MeV nn pp 07/08/2019 8 R. Lazauskas 8

Complex scaling method (2b scattering) nn pp 07/08/2019 9 R. Lazauskas 9

Complex scaling method (2b scattering) N-12C at Elab=30 MeV (2S1) 07/08/2019 10 R. Lazauskas 10

Faddeev eq. (particles of identical mass) Complex scaling method (3b scattering) Faddeev eq. (particles of identical mass) x1 y1 y2 y3 x3 1 2 3 x2 Outgoing 2b or 3b wave 07/08/2019 11 R. Lazauskas 11

Faddeev eq. (particles of identical mass) Complex scaling method (3b scattering) Faddeev eq. (particles of identical mass) x1 y1 y2 y3 x3 1 2 3 x2 Complex scaling Outgoing wave becomes exponentialy bound: 2-body plane out. waves 3-body break-up out. wave 07/08/2019 12 R. Lazauskas 12

Faddeev eq. (particles of identical mass) Complex scaling method (3b scattering) Faddeev eq. (particles of identical mass) x1 y1 y2 y3 x3 1 2 3 x2 Complex scaling Ensure that the terms are bound!! y2 Domain of interaction ? Domain of resolution Angle q becomes small for large scattering energies Elab OK x2 07/08/2019 13 R. Lazauskas 13

Complex scaling method (3b scattering) Extraction of the scattering amplitudes: From the asymptote of the solution Using Green’s theorem y1 Domain of interaction Domain of resolution Cumbersome for break-up!! OK x1 07/08/2019 14 R. Lazauskas 14

Complex scaling method (3b scattering) Ref.[23] J. L. Friar et al.:, Phys. Rev. C 51 (1995) 2356. Ref.[24] A. Deltuva, A. C. Fonseca et al.:,, Phys. Rev. C 71 (2005) 064003. 07/08/2019 15 R. Lazauskas 15

Complex scaling method (3b scattering) Ref.[24] A. Deltuva, A. C. Fonseca et al.:, Phys. Rev. C 71 (2005) 064003. 07/08/2019 16 R. Lazauskas 16

Complex scaling method (3b scattering) Ref.[23] J. L. Friar et al.:, Phys. Rev. C 51 (1995) 2356. 07/08/2019 17 R. Lazauskas 17

Complex scaling method AV18 Optical CH89 pot. 12C n (Deltuva) (Deltuva) 07/08/2019 18 R. Lazauskas 18

Conclusion R. Lazauskas Complex scaling method is efficient tool to solve bound, resonant as well as continuum states problem without explicit treatment of the boundary conditions Scattering problem might be solved using bound state methods in complex arithmetic Simple extension of the formalism to many-body scattering case Reliable results are already obtained for 3-body elastic and break-up scattering, including long-range and optical potentials 07/08/2019 19 R. Lazauskas 19