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 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.

34 pages, 1 figure, accepted for publication in Phys. Rev. A

References in corpus (4)

• Transfer matrix in scattering theory: A survey of basic properties and recent developments
• A Class of Exactly Solvable Scattering Potentials in Two Dimensions, Entangled State Pair Generation, and a Grazing Angle Resonance Effect
• Transfer matrix for long-range potentials
• Low-frequency scattering defined by the Helmholtz equation in one dimension

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• Dynamical formulation of low-frequency scattering in two and three dimensions
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• Singularity-free treatment of delta-function point scatterers in two dimensions and its conceptual implications
• Propagating-wave approximation in two-dimensional potential scattering
• Reciprocity Theorem and Fundamental Transfer Matrix
• Low-Frequency Scattering of TE and TM Waves by an Inhomogeneous Medium with Planar Symmetry
• Renormalization of multi-delta-function point scatterers in two and three dimensions, the coincidence-limit problem, and its resolution
• Scattering of TE and TM waves by inhomogeneities of a 2D material, low-frequency behavior of the scattering amplitude, and low-frequency invisibility
• Existence of the transfer matrix for a class of nonlocal potentials in two dimensions
• Introducing a general method for solving electromagnetic radiation problem in an arbitrary linear medium