Wave-packet continuum discretization for quantum scattering

Abstract

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-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 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.

Quantum Scattering Theory in a Discrete Representation

“…In fact, one can build the similar wave packets for a Hamiltonian which includes the long-range Coulomb interaction, for example, and then derive an analytical finite-dimensional representation for the Coulomb resolvent [4] as well.…”

“…Then the complete system of eigenfunctions for the Hamiltonian h is constructed from its bound states |ψ n and eigendifferentials (see the details in ref. [4]). The Hamiltonian matrix h and also the matrix of the resolvent g(E) = [E + i0 − h] −1 for the Hamiltonian h have explicit diagonal forms in a WP representation.…”

“…As an appropriate basis set we have taken the stationary wave packets, which have been introduced by Herman Weyl under a title of eigendifferentials [3]. The detailed description of the whole wave-packet approach and the main results attained can be found in our very recent review paper [4]. Here we briefly outline the basic results related to a construction of a stationary wave-packet basis in few-body case, solving the realistic 3N scattering problem on ordinary PCs using the Graphics Processing Unit (GPU) technique.…”

“…Nearly a decade ago, our group in Moscow State University have developed an original approach based on the complete few-body continuum discretization and formulation of the problem in a finite L 2 basis which corresponds to the treatment of a scattering problem in a discrete representation [4][5][6]. As a result of such a projection one arrives at a fully algebraic scheme for solving the initial problem in which all the operators and wave functions are represented by matrices and vectors, while the scattering equations are reduced to matrix ones.…”

“…{23}1) with momenta (p, q), where h 0p and h 0q are kinetic energy operators and v 1 is the interaction between particles 2 and 3. Then one can introduce free WP states |X i j and channel WP states |Z k j from two-body ones (with account of necessary spin-angular couplings) and also connect the latter with the former by a simple rotation [4] (all the necessary quantum numbers have to be taken into account) 2 . One of the main advantages of the discrete WP approach is that the scattering problem can be solved directly in the basis space corresponding to the channel Hamiltonian instead of the three-body free Hamiltonian space.…”