O.A. Rubtsova, V.I. Kukulin, V.N. Pomerantsev O.A. Rubtsova, V.I. Kukulin, V.N. Pomerantsev Institute of Nuclear Physics, Moscow State University Institute of Nuclear Physics, Moscow State University

19th International IUPAP Conference on Few-Body Problems in Physics

31 August 2009

Solving Solving FFew-ew-BBody ody SScattering cattering PProblemsroblems in the in the MMomentum omentum LLattice attice BBasisasis

OutlineOutline1. Motivation2. Wave-packet formalism Stationary wave packets and their properties Construction of scattering wave-packets as pseudostates Discretization procedure Finite-dimensional representations for basic scattering operators3. Multi-channel scattering problem Solution via matrix equations New treatment of the multi-channel pseudostates4. Three- and few-body scattering problem Channel wave-packet bases Solution of Faddeev-like equations in the lattice basis5. Solving scattering problems without equations Discrete spectral shift formalism6. Summary

MotivationMotivation 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 discretization with an L2 basis is preferable for few-body scattering calculations.

Stationary wave packets Stationary wave packets —— the lattice basis the lattice basisDiscretization of the free Hamiltonian H0 continuum

E0 E1 Ei-1 Ei EN

i

Behavior of basis functions

In the coordinate space

WP functions have very long-range behavior inthe coordinate space, while in the momentum space they are represented by step-functions

Wave packets as a basis: Wave packets as a basis:

approximation of bound- and pseudo- statesapproximation 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.

Discretization procedureDiscretization procedure The wave-packet continuum discretization procedure consists of three steps.

2. Finite-dimensional representations for scattering operators:

1.

3.

Discrete version of the quantum scattering theoryDiscrete version of the quantum scattering theoryFinite-dimensional analogs of the basic scattering theory operators

Free resolvent Transition operator

WP calculations with the non-local potentialWP calculations with the non-local potential

V(r,r')= — U(|r+r'|)W(|r-r'|)

n+Fe differential cross section at En=7 MeV

S-wave phase shift and inelasticity

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.

• N=5— N=10— N=20— N=40

Convergence in WP basis

Multi-channel scattering problemMulti-channel scattering problem

The general integral wave-packet formalism remains the same as in the one-channel case:

T=V+VG0 T

Scattering observables can be found from the matrix analog of the multi-channel Lippmann-Schwinger equation for T-matrix:

Approximation for the total Approximation for the total resolventresolventvia pseudostatesvia pseudostates

Model two-channel problem:Weights of pseudostates of total

Hamiltonian

H0

H0

The main problem is how to approximate multi-channel scattering WPs via diagonalization of the total Hamiltonain matrix

H0

In a finite-dimensional basis,

the spectrum of multi-channel

Hamiltonian is not degenerated.

with multi-channel scattering wave-packets

weights of the 1st channel component of pseudostates

One should use degenerate discretized spectrum of the free Hamiltonian to

distinguish total Hamiltonian pseudostates

H0

splitting

H H0 H0

1 2

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.

w1even + w1

odd = 1

In each pair of splitted states, weights of channel components related to each other:

NN interaction with the tensor force (Moscow potential)

s-wave phase shift d-wave phase shift

mixing parameter weights for the ‘even’ and ‘odd’ pseudostates

Three- and few-body problemThree- 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:

Formulation of three-body problem in theFormulation of three-body problem in the Faddeev-like lattice frameworkFaddeev-like lattice framework

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.

these m.e. are independent on energy

and interaction

Elastic n-d scatteringElastic n-d scattering

Real part of the S-wave phase shift (quartet channel)

Inelasticity (quartet channel)

Calculations with local MT NN interaction

(V.N. Pomerantsev, V.I. Kukulin, O.A. Rubtsova, PRC 79, 034001 (2009))

N=200x200 N=100x100

standard Faddeev calc. (J.L. Friar at al., Phys.Rev. C 42, 1838 (1990))

Real part of the S-wave phase shift (doublet channel)

Inelasticity (doublet channel)

N=(100+100)x100 N=(50+50)x50

standard Faddeev calc.

Solution of multi-channel scattering problems without equationsSolution of multi-channel scattering problems without equationsV.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

This formalism is valid also for a complex potential

Thus, the solution of the scattering problem is reduced to diagonalization of the total Hamiltonian matrix in the lattice basis

Mutli-channel calculations

channel phase shifts and mixing parameter

elastic and reaction cross section in the channel 1

e

r

Channel phase shift for multi-channel Hamiltonian can be found just from the difference between the pseudostate and the free Hamiltonian eigenvalues

Discrete analog for the spectral shift function

In the Discrete spectral shift formalism, one canobtain 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.

Two-channel model potential for e-H scattering

B.H. Bransden and A.T. Stelbovics, J. Phys. B: At. Mol. Phys. 17, 1877 (1984).

ConclusionConclusion

(i) The explicit analytical f.-d. representations for channel resolvents allow (i) The explicit analytical f.-d. representations for channel resolvents allow to reduce initial integral equations to the matrix ones those can be to reduce initial integral equations to the matrix ones those can be solved directly on the real energy axis.solved directly on the real energy axis.

(ii) The very long-range type of the wave-packet functions allows to (ii) The very long-range type of the wave-packet functions allows to approximate properly the overlapping between WF components in approximate properly the overlapping between WF components in different Jacoby coordinate sets. different Jacoby coordinate sets.

(iii) The matrix representation for interaction operators allows to work with (iii) The matrix representation for interaction operators allows to work with non-local potentials in the same numerical scheme as for local ones.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.(iv) The lattice basis can also be used to study few-body resonance states.

(v) Developed formalism allows to construct effective optical-model (v) Developed formalism allows to construct effective optical-model potentials of composite particle interaction.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.

Composite particle scattering on a heavy targetComposite particle scattering on a heavy target

R

r

b

c

A

Elastic d+Elastic d+5858Ni scattering: calculations within the WPCD and CDCC methodsNi scattering: calculations within the WPCD and CDCC methods

Study of a close channel influence on the elastic scattering amplitude

d+ Ni elastic c.s. at Ed=21.6 MeV 58

(converged — Emax= 110 MeV)

(O.A. Rubtsova, V.I. Kukulin, A.M.M. Moro, PRC 78, 034603 (2008))

n-p spectrum

E

Emax

open channels

closed channels

0

In the WP approach, closed and open channels are treated jointly.

d+ Ni elastic c.s. at Ed=12 MeV

converged — Emax= 77 MeV

58

Finite-dimensional approximation for the channel resolventFinite-dimensional approximation for the channel resolvent

Construction of effective optical potentials in the Construction of effective optical potentials in the wave-packet basiswave-packet basis

Feshbach projection technique for {bc}+A system Wave-packet approximation

Feshbach potential for d+Ni problem Feshbach potential for d+Ni problem at Eat Edd=80 MeV=80 MeV

Real Real partpart

Imaginary Imaginary partpart

L=0