Wave-packet continuum discretization for quantum scattering

TL;DR

This paper introduces a wave-packet continuum discretization method for quantum scattering, transforming continuous spectrum problems into discrete linear algebra problems, simplifying complex few- and many-body scattering calculations.

Contribution

It presents a novel discretization approach using stationary wave packets, enabling fully discrete treatment of complex scattering problems and finite-dimensional operator approximations.

Findings

Discretization simplifies solving few- and many-body scattering problems.

Finite-dimensional approximations for complex operators are achieved.

Method applied to multichannel and three-nucleon scattering problems.

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.