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O.A. Rubtsova, V.I. Kukulin, V.N. Pomerantsev

Solving F ew- B ody S cattering P roblems in the M omentum L attice B asis. O.A. Rubtsova, V.I. Kukulin, V.N. Pomerantsev Institute of Nuclear Physics, Moscow State University. 19th International IUPAP Conference on Few-Body Problems in Physics 31 August 2009. Outline.

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O.A. Rubtsova, V.I. Kukulin, V.N. Pomerantsev

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  1. Solving Few-Body Scattering Problems in the Momentum Lattice Basis O.A. Rubtsova, V.I. Kukulin, V.N. Pomerantsev Institute of Nuclear Physics, Moscow State University 19th International IUPAP Conference on Few-Body Problems in Physics 31 August 2009

  2. Outline 1. Motivation 2. Wave-packet formalism Stationary wave packets and their properties Construction of scattering wave-packets as pseudostates Discretization procedure Finite-dimensional representations for basic scattering operators 3. Multi-channel scattering problem Solution via matrix equations New treatment of the multi-channel pseudostates 4. Three- and few-body scattering problem Channel wave-packet bases Solution of Faddeev-like equations in the lattice basis 5. Solving scattering problems without equations Discrete spectral shift formalism 6. Summary

  3. Motivation • Direct solution of few-body scattering problems via integral equations is rather cumbersome numerically and leads to time-consuming procedures, especially in charged particle case and when modern interactions such as 3B forces are included. L2-type techniques are very effective in few-body bound-state calculations. • Implementation of L2-type methods and partial continuum discretization have been shown to provide convenient calculation techniques for some partial three- and few-body scattering problems: • * scattering of composite particles off nuclei (the CDCC technique); • * true many-particle scattering (solution in the Harmonic Oscillator Representation); • * photo-disintegration of light nuclei (Lorentz Integral Transform method); • and other approaches. • Discretization of continuous spectrum should be used to prove some basic properties of operators in Hilbert space. That is why continuum discretizationwith an L2 basis is preferable for few-body scattering calculations.

  4. Stationary wave packets — the lattice basis Behavior of basis functions Discretization of the free Hamiltonian H0 continuum i In the coordinate space E0 E1 Ei-1 Ei EN WP functions have very long-range behavior in the coordinate space, while in the momentum space they are represented by step-functions

  5. Wave packets as a basis: approximation of bound- and pseudo- states Diagonalization procedure: Pseudostates of any Hamiltonian H constructed in free wave-packet basis can be interpreted as scattering WPs which correspond to scattering wave-functions of H. For example, Coulomb wave-packets can be constructed on the finite free-packet basis via the diagonalization procedure, while exact regular Coulomb wave-functions cannot be expanded on the plane-wave set.

  6. Discretization procedure The wave-packet continuum discretization procedure consists of three steps. 1. 2. Finite-dimensional representations for scattering operators: 3.

  7. Discrete version of the quantum scattering theory Finite-dimensional analogs of the basic scattering theory operators Free resolvent Transition operator

  8. WP calculations with the non-local potential V(r,r')= — U(|r+r'|)W(|r-r'|) S-wave phase shift and inelasticity Convergence in WP basis n+Fe differential cross section at En=7 MeV • N=5 — N=10 — N=20 — N=40 The results of the WP method for the non-local potential are in very good agreement with results of a direct numerical solution for the Local Phase-Equivalent potential at different energies.

  9. Multi-channel scattering problem The general integral wave-packet formalism remains the same as in the one-channel case: Scattering observables can be found from the matrix analog of the multi-channel Lippmann-Schwinger equation for T-matrix: T=V+VG0 T

  10. Approximation for the total resolvent via pseudostates a b H0 H0 In a finite-dimensional basis, the spectrum of multi-channel Hamiltonian is not degenerated. H0 with multi-channel scattering wave-packets Model two-channel problem: Weights of pseudostates of total Hamiltonian The main problem is how to approximate multi-channel scattering WPs via diagonalization of the total Hamiltonain matrix

  11. One should use degenerate discretized spectrum of the free Hamiltonian to distinguish total Hamiltonian pseudostates weights of the 1st channel component of pseudostates splitting In each pair of splitted states, weights of channel components related to each other: 2 1 H0 H0 H H0 w1even + w1odd = 1 Although spectrum of the total Hamiltonian is not degenerated in the lattice basis, we have series of states at each discrete energy which can be considered as wave-packet analogs of scattering states corresponding to different initial boundary conditions.

  12. NN interaction with the tensor force (Moscow potential) weights for the ‘even’ and ‘odd’ pseudostates mixing parameter d-wave phase shift s-wave phase shift

  13. Three- and few-body problem In general few-body case, WP basis should be constructed in each Jacobi coordinate set via direct production of each subsystem WP bases. Such a basis consists of eigenstates of each channel Hamiltonian. Lattice representation leads to a complete few-body continuum discretization. The main advantage here is finite-dimensional representation for the few-body channel resolvent:

  14. Formulation of three-body problem in the Faddeev-like lattice framework these m.e. are independent on energy and interaction Thus, the Lattice Representation for channel resolvents and components of WF allows to find solution of general thee- (or few-) body scattering problem in a three-(or few-) body L2 WP basis.

  15. Elastic n-d scattering (V.N. Pomerantsev, V.I. Kukulin, O.A. Rubtsova, PRC 79, 034001 (2009)) Real part of the S-wave phase shift (quartet channel) Inelasticity (quartet channel) N=200x200 N=100x100 Calculations with local MT NN interaction standard Faddeev calc. (J.L. Friar at al., Phys.Rev. C 42, 1838 (1990))

  16. Real part of the S-wave phase shift (doublet channel) Inelasticity (doublet channel) N=(100+100)x100 N=(50+50)x50 standard Faddeev calc.

  17. Solution of multi-channel scattering problems without equations V.N. Pomerantsev, V.I. Kukulin, O.A. Rubtsova, JETP letters 90, 443 (2009). Discrete spectral shift formalism Based on the Lifshitz—Birman—Krein Spectral Shift Function formalism Thus, the solution of the scattering problem is reduced to diagonalization of the total Hamiltonian matrix in the lattice basis This formalism is valid also for a complex potential

  18. Two-channel model potential for e-H scattering B.H. Bransden and A.T. Stelbovics, J. Phys. B: At. Mol. Phys. 17, 1877 (1984). Mutli-channel calculations Discrete analog for the spectral shift function channel phase shifts and mixing parameter Channel phase shift for multi-channel Hamiltonian can be found just from the difference between the pseudostate and the free Hamiltonian eigenvalues elastic and reaction cross section in the channel 1 In the Discrete spectral shift formalism, one can obtain all multi-channel S-matrix elements for a wide energy region via a single total Hamiltonian matrix diagonalization in the WP basis — WP basis • conventional calc. se sr

  19. Conclusion (i) The explicit analytical f.-d. representations for channel resolvents allow to reduce initial integral equations to the matrix ones those can be solved directly on the real energy axis. (ii) The very long-range type of the wave-packet functions allows to approximate properly the overlapping between WF components in different Jacoby coordinate sets. (iii) The matrix representation for interaction operators allows to work with non-local potentials in the same numerical scheme as for local ones. (iv) The lattice basis can also be used to study few-body resonance states. (v) Developed formalism allows to construct effective optical-model potentials of composite particle interaction. Thus, the developed Lattice Technique should be considered as a convenient new language of discretized calculations in the few-body scattering theory.

  20. Composite particle scattering on a heavy target b R A r c

  21. Elastic d+58Ni scattering: calculations within the WPCD and CDCC methods (O.A. Rubtsova, V.I. Kukulin, A.M.M. Moro, PRC 78, 034603 (2008)) Study of a close channel influence on the elastic scattering amplitude 58 d+ Ni elastic c.s. at Ed=21.6 MeV (converged — Emax= 110 MeV) n-p spectrum Emax closed channels E In the WP approach, closed and open channels are treated jointly. open channels 0

  22. 58 d+ Ni elastic c.s. at Ed=12 MeV converged — Emax= 77 MeV

  23. Finite-dimensional approximation for the channel resolvent

  24. Construction of effective optical potentials in the wave-packet basis Feshbach projection technique for {bc}+A system Wave-packet approximation

  25. Feshbach potential for d+Ni problem at Ed=80 MeV Real part L=0 Imaginary part

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