Wave-packet continuum discretization for quantum scattering

A general approach to a solution of few- and many-body scattering problems based on a continuum-discretization procedure is described in detail. The complete discretization of continuous spectrum is realized using stationary wave packets which are the normalized states constructed from exact non-normalized continuum states. Projecting the wave functions and all scattering operators like t$t$-matrix, resolvent, etc. on such a wave-packet basis results in a formulation of quantum scattering problem entirely in terms of discrete elements and linear equations with regular matrices. It is demonstrated that there is a close relation between the above stationary wave packets and pseudostates which are employed often to approximate the scattering states with a finite L₂$L_2$ basis. Such a fully discrete treatment of complicated few- and many-body scattering problems leads to significant simplification of their practical solution. Also we get finite-dimensional approximations for complicated operators like effective interactions between composite particles constructed via the Feshbach-type projection formalism. As illustrations to this general approach we consider several important particular problems including multichannel scattering and scattering in the three-nucleon system within the Faddeev framework.     

New approach to solving the problem of composite-particle scattering on nuclei

An essentially new approach to solving the problem of elastic and inelastic scattering of a composite particle on stable nuclei is described. Within this approach, all channels of virtual breakup and stripping in the intermediate states are included in a nonlocal complex-valued interaction operator with the aid of the projection-operator technique.The three-particle continuum spectrum of the Hamiltonian for intermediate states in Q space is calculated within the orthogonalizing-pseudopotential method by introducing a pseudo-Hamiltonian, which is diagonalized in a full space in terms of a relevant oscillator basis. As was shown by a number of authors, the use of special quadratures makes it possible to reduce integration over the continuous spectrum of intermediate states to summation over a discretized continuum. On the basis of the formalism developed in this study, a closed Schrödinger equation with a nonlocal complex potential for partial waves is derived for describing elastic scattering of a composite particle by a target, and an explicit approximate formula for the amplitude of three-particle breakup is obtained on the same basis. This method has a number of obvious advantages over currently well-known approaches of the type of the discretized-continuum coupled-channel method, where solving the problem in question reduces to solving a cumbersome set of coupled equations.     

Effective Field Theory and the Gamow Shell Model

We combine halo/cluster effective field theory (H/CEFT) and the Gamow shell model (GSM) to describe the 0⁺ ground state of ⁶He as a three-body halo system. We use two-body interactions for the neutron-alpha particle and two-neutron pairs obtained from H/CEFT at leading order, with parameters determined from scattering in the p_3/2 and s₀ channels, respectively. The three-body dynamics of the system is solved using the GSM formalism, where the continuum states are incorporated in the shell model valence space. We find that in the absence of three-body forces the system collapses, since the binding energy of the ground state diverges as cutoffs are increased. We show that addition at leading order of a three-body force with a single parameter is sufficient for proper renormalization and to fix the binding energy to its experimental value.     

Coupling effects of resonant and discretized non-resonant continuum states in4He +6Li scattering at 10 MeV/A

Alpha- particle scattering from the resonant (3 ₁ ⁺ ) and non-resonant continuum states of⁶Li is studied at incident energy 10 MeV/A. Theα+d breakup continuum part within the excitation energyE _ex=1.475–2.475 MeV is discretized in two energy bins. Unlike the results at higher incident energies, here the coupled-channel calculations show significant breakup continuum coupling effects on the elastic and inelastic scattering. It is shown that even when the continuum-continuum coupling effects are strong, the experimental data of the ground state and the resonant as well as discretized non-resonant continuum states impose stringent constraint on the coupling strengths of the non-resonant continuum states.     

Quantum scattering theory on the momentum lattice

Here we briefly outline main results of the Wave-Packet Continuum Discretization method. The formalism uses the complete continuum discretization scheme in terms of the momentum stationary wave-packet basis, which leads to formulation of the scattering problem on a lattice in the momentum space.     

Dynamic polarization of 6,7Li nuclei in the elastic scattering of α particles

The elastic scattering of α particles on weakly bound ^6,7Li cluster nuclei is considered with allowance for their dynamic polarization within the three-particle model. The considered states of the αd and αt continuums are projected on a finite basis of stationary wave packets, which allows the total three-body problem to be reduced to a matrix problem. The results from calculating α + ^6,7Li-elastic scattering differential cross sections are considered as illustration of the three-particle approach and is compared to the results of other authors.     

Borromean three-body heteroatomic resonances

We present an approach to analyze recent experimental evidences of Efimov resonant states in mixtures of ultracold gases, by considering two-species three-body atomic systems bound in a Borromean configuration, where all the two-body interactions are unbound. For such Borromean three-body systems, it is shown that a continuum three-body s-wave resonance emerges from an Efimov state as a scattering length or a three-body scale is moved. The energy and width of the resonant state are determined from a scaling function with arguments given by dimension-less energy ratios relating the two-body virtual state subsystem energies with the shallowest three-body bound state. The peculiar behavior of such resonances is that their peaks are expected to move to lower values of the scattering length, with increasing width, as one raises the temperature. For Borromean systems, two resonant peaks are expected in ultralow-temperature regimes, which will disappear at higher energies. It is shown how a Borromean-Efimov excited bound state turns out to a resonant state by tuning the virtual two-body subsystem energies or scattering lengths, with all energies written in units of the next deeper shallowest Efimov state energy. The resonance position and width for the decay into the continuum are obtained as universal scaling functions (limit cycle) of the dimensionless ratios of the two and three-body scales, which are calculated numerically within a zero-range renormalized three-body model.     

Alternative treatment of the three-body problem

For solving the three-body problem with local potentials a model HamiltonianH ₀ containing an interaction between one particle and the centre-of-mass of the other two interacting particles is introduced. The total HamiltonianH is obtained byH=H _o +W whereW is a “residual interaction” in close analogy to the nuclear shell model. At a certain stage of the calculationsH ₀ has to be replaced by a new model Hamiltonian \(\tilde H_0 \) containing plane waves. The resolvent (and thereby theT-matrix) of the three-body problem is calculated by operator techniques. It is possible to draw some conclusions concerning three-body properties from these general expressions. Therefore this attempt may be considered as a supplementary treatment, in addition to the Faddeev-equations, of the three-body problem: it exhibits the discrete spectrum, the simple and the twofold continuum ofH arising from the corresponding states ofH ₀, and provides some approximation methods.     

Structure and dynamics of few-helium clusters using soft-core potentials

In this work we investigate the structure and dynamics of small clusters of Helium atoms. We consider bound states of clusters having A = 2, 3, 4, 5, 6 atoms and continuum states in the three-atom system. Motivated by the fact that the He-He system has a very large scattering length a compared to the range r ₀ of the He-He potential (r ₀/a < 1/10), we propose the use of a soft-core interparticle potential. We use an attractive gaussian potential that reproduces the values of the dimer binding energy and the atomatom scattering length obtained with one of the widely used He-He interactions, the LM2M2 potential. In addition, we include a repulsive three-body force to reproduce the trimer binding energy. With this model, consisting in the sum of a two- and a three-body potential, we show the spectrum of the four, five, and sixparticle systems and phase-shifts and inelasticities in the three-atom system. Comparisons to calculations using realistic He-He potentials are given. In addition some universal relations are explored.     

Tests of the discretized-continuum method in three-body dipole strengths

We investigate the ⁶He dipole distribution in a three-body α+n+n model. Two approaches are used to describe the three-body ¹⁻ continuum: the discretized-continuum method, where the scattering wave functions are approximated by square-integrable functions, and the R-matrix formalism, where their asymptotic behaviour is taken into account. We show that some ambiguity exists in the pseudostate method, owing to the smoothing technique, necessary to derive continuous distributions. We show evidence for the important role of the halo structure in the E1 dipole strength. We also address the treatment of Pauli forbidden states in the three-body wave functions.     

Interband transitions and electronic Raman continuum in systems with strong inelastic scattering: Applications to YBa2Cu3O7-δ

We consider a model for the electronic Raman continuum which takes into account strong inelastic scattering and interband transitions. Calculations are based on four-vertex Raman scattering diagrams (Kawabata formalism) within the RPA for Coulomb interaction and the ladder diagram Bethe-Salpeter equation for the vertex. We apply this method to an analysis of the nature of the electronic Raman continuum in the normal state of the high-T _c superconductor YBa₂Cu₃O₇. In numerical calculations we take into account all the self-energy effects and make simulations for vertex corrections assuming that inelastic scattering is due to electron-phonon interaction. Theab-plane polarized continuum contains a large contribution from interband processes and does not depend strongly on temperature and inelastic scattering strength. The in-plane anisotropy is determined by interband transitions rather than by anisotropy of the Fermi surface. The ZZ continuum contains only weak contribution from interband transitions. It can be crudely described within a single band model with inelastic scattering and is strongly dependent on the relaxation rates of inelastic scattering. The nature of the oxygen-deficiency dependence of the Raman spectra is also commented upon.     

Precise numerical results for limit cycles in the quantum three-body problem

The study of the three-body problem with short-range attractive two-body forces has a rich history going back to the 1930s. Recent applications of effective field theory methods to atomic and nuclear physics have produced a much improved understanding of this problem, and we elucidate some of the issues using renormalization group ideas applied to precise nonperturbative calculations. These calculations provide 11-12 digits of precision for the binding energies in the infinite cutoff limit. The method starts with this limit as an approximation to an effective theory and allows cutoff dependence to be systematically computed as an expansion in powers of inverse cutoffs and logarithms of the cutoff. Renormalization of three-body bound states requires a short range three-body interaction, with a coupling that is governed by a precisely mapped limit cycle of the renormalization group. Additional three-body irrelevant interactions must be determined to control subleading dependence on the cutoff and this control is essential for an effective field theory since the continuum limit is not likely to match physical systems (e.g., few-nucleon bound and scattering states at low energy). Leading order calculations precise to 11-12 digits allow clear identification of subleading corrections, but these corrections have not been computed.     

Microscopic theory of nuclear reactions based on the wave-packet technique

A new approach to describe the dynamics of nuclear reactions is proposed. The approach is based on the discretization of the total Hamiltonian continuum. The usage of the special stationary wave-packet basis makes it possible to reduce the scattering integral equations to matrix ones. The energy singularities of the initial equation kernels are smoothed by the averaging over discretization intervals. The approach is highly universal and allows one to properly take into account various exchange processes. In addition, the wave-packet technique makes it possible to construct effective optical potentials of interaction between composite nuclei.     

Coupling effects of resonant and discretized non-resonant continuum states in 4He + 6Li scattering at 10 MeV/A

Alpha-particle scattering from the resonant (3 ₁ ⁺ ) and non-resonant continuum states of ⁶Li is studied at incident energy 10 MeV/A. The α + d breakup continuum part within the excitation energy E _ex = 1.475–2.475 MeV is discretized in two energy bins. Unlike the results at higher incident energies, here the coupled-channel calculations show significant breakup continuum coupling effects on the elastic and inelastic scattering. It is shown that even when the continuum-continuum coupling effects are strong, the experimental data of the ground state and the resonant as well as discretized non-resonant continuum states impose stringent constraint on the coupling strengths of the non-resonant continuum states.     

Elastic scattering, muon transfer, bound states and resonances in the three-body mesic molecular systems in the reduced adiabatic hyperspherical approach

The uniform method of numerical investigation of bound states and scattering processes 2→ 2 (including resonance states) in the Coulomb three-body (CTB) systems is developed. It is based on the adiabatic hyperspherical approach (AHSA) and includes the numerical realization and applications to the three-body mesic atomic systems. The results of calculations of bound states of these systems (including the local characteristics of the wave functions) and the scattering processes 2→ 2 (including the characteristics of the resonance states) are presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

Asymptotic method for determining the amplitude for three-particle breakup: Neutron-deuteron scattering

The process of neutron-deuteron scattering at energies above the deuteron-breakup threshold is described within the three-body formalism of Faddeev equations. Use is made of the method of solving Faddeev equations in configuration space on the basis of expanding wave-function components in the asymptotic region in bases of eigenfunctions of specially chosen operators. Asymptotically, wave-function components are represented in the form of an expansion in an orthonormalized basis of functions depending on the hyperangle. This basis makes it possible to orthogonalize the contributions of elastic-scattering and breakup channels. The proposed method permits determining scattering and breakup parameters from the asymptotic representation of the wave function without reconstructing it over the entire configuration space. The scattering and breakup amplitudes for states of total spin S = 1/2 and 3/2 were obtained for the s-wave Faddeev equation.     

SS-HORSE Method for Analysis of Resonances: Charged-Particle Scattering

The method for analyzing resonance states based on the harmoni©oscillator representation of scattering equations (HORSE) and on analytic properties of partial-wave scattering amplitudes which has been proposed earlier, is generalized to the case of charged-particle scattering. The method in question is tested by applying it to the model problem of pα scattering and can be used to study resonance states on the basis of microscopic calculations within various versions of the shell model.     

Solving the relativistic inverse scattering problem with allowance for inelasticity effects on the basis of <Emphasis Type="Italic">N/D</Emphasis> equations and application of the resulting solution to an analysis of nucleon-nucleon interaction

A manifestly Poincaré-invariant approach to solving the inverse scattering problem is developed with allowance for inelasticity effects. The equations of the N/D method are used as dynamical equations in this approach. Two versions of the approach are considered. In the first version (method A), the required equations are constructed on the basis of the maximal-analyticity principle, which constitutes the basis of dynamical S-matrix theory. In formulating the second version of equations (method B), it is assumed that a partial-wave scattering amplitude may develop dynamical singularities that violate the requirement of maximal analyticity. The dynamics of interaction components that violate maximal analyticity is described within the model of a nonlocal separable potential. The method is used to analyze nucleon-nucleon interaction in the ¹S₀ and ³S₁ states. The results obtained by solving the inverse scattering problem for potential functions are compared with the predictions of the one-boson-exchange model.