Fundamental transfer matrix and dynamical formulation of stationary scattering in two and three dimensions

TL;DR

This paper develops a multidimensional transfer matrix framework for stationary scattering in 2D and 3D, improving previous models by incorporating evanescent waves and analyzing special potentials.

Contribution

It introduces a consistent dynamical formulation using operator-valued transfer matrices, extending the concept to higher dimensions and clarifying their properties and applications.

Findings

Provides a new transfer matrix formulation for 2D and 3D scattering.

Shows how to handle delta-function potentials and their regularization.

Characterizes invisible and exactly solvable potentials.

Abstract

We offer a consistent dynamical formulation of stationary scattering in two and three dimensions that is based on a suitable multidimensional generalization of the transfer matrix. This is a linear operator acting in an infinite-dimensional function space which we can represent as a $2\times 2$ matrix with operator entries. This operator encodes the information about the scattering properties of the potential and enjoys an analog of the composition property of its one-dimensional ancestor. Our results improve an earlier attempt in this direction [Phys. Rev. A 93, 042707 (2016)] by elucidating the role of the evanescent waves. In particular, we show that a proper formulation of this approach requires the introduction of a pair of intertwined transfer matrices each related to the time-evolution operator for an effective non-unitary quantum system. We study the application of our findings in the treatment of the scattering problem for delta-function potentials in two and three dimensions and clarify its implicit regularization property which circumvents the singular terms appearing in the standard treatments of these potentials. We also discuss the utility of our approach in characterizing invisible (scattering-free) potentials and potentials for which the first Born approximation provides the exact expression for the scattering amplitude.