1. R. Lazauskas Application of the complex-scaling
method for few-body scattering
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2. 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…
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3. 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
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4. 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:
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5. 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)
R. Lazauskas and J. Carbonell, Phys. Rev. C 72 (2005)
R. Lazauskas and J. Carbonell, Phys. Rev. C 71, (2005)
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6. Complex scaling method (Resonances)
Analiticity of the potential, limitation of the scaling angle:
Starts to diverge for q>p/(2n) 6
Hardcore propagates!
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7. 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
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8. Complex scaling method (2b scattering)
Ecm=1 MeV nn pp
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9. Complex scaling method (2b scattering)nn pp
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10. Complex scaling method (2b scattering)
N-12C at Elab=30 MeV (2S1)
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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
Outgoing
2b or 3b wave
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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
Outgoing wave becomes exponentialy bound:
2-body plane out. waves
3-body break-up out. wave
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13. 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
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14. 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
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15. 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)
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16. Complex scaling method (3b scattering)
Ref.[24] A. Deltuva, A. C. Fonseca et al.:, Phys. Rev. C 71 (2005)
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17. Complex scaling method (3b scattering)
Ref.[23] J. L. Friar et al.:, Phys. Rev. C 51 (1995) 2356.
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18. Complex scaling method
AV18
Optical CH89 pot.
12C n
(Deltuva)
(Deltuva)
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19. 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
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