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

Abstract

We offer a consistent dynamical formulation of stationary scattering in two and three dimensions (2D and 3D) 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 × 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. 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 nonunitary quantum system. We study the application of our findings in the treatment of the scattering problem for δ-function potentials in 2D and 3D 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.

Transfer matrix formulation of stationary scattering in 2D and 3D: A concise review of recent developments

“…If v(x, y) vanishes for a range of values of x, V (x) = 0 and H(x) = 0. This feature of H(x) implies the composition property of the (auxiliary) transfer matrix [9]. As we explain in Ref.…”

“…They involved a discretization of all but one of degrees of freedom and produced large numerical transfer matrices with a build-in composition property which allowed for numerical treatment of wave propagation and scattering. The developments reported in [8,9] are of a completely different nature, for they introduce a fundamental notion of the transfer matrix which is amenable to analytic calculations.…”

“…The fundamental transfer matrix enjoys a composition property similar to its one-dimensional analogue. The derivation of this property requires the use of a related object called the "auxiliary transfer matrix" [9]. This is defined as the 2 × 2 matrix M with operator entries…”

“…Refs. [8,9] develop a higher-dimensional generalization of the DFSS where the role of the transfer matrix M is played by a linear operator acting in an infinite-dimensional function space. This operator, which we also call the "transfer matrix," has a canonical realization as a 2 × 2 matrix M with operator entries M ij .…”

“…One of the most remarkable benefits of DFSS in two and three dimensions is that its application to delta-function potentials in these dimensions [8,9] circumvents the unwanted singularities of their standard treatments [10,11]. Among other applications of this approach to scattering theory are the discovery of a class of invisible (scattering-free) complex potentials [9,12,13,14,15] in two and three-dimensions and the construction of their electromagnetic counterparts [16].…”