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arXiv:2008.05542 · Physical Review A

Symmetry-induced nonlocal divergence-free currents in two-dimensional quantum scattering

M. A. Metaxas, Peter Schmelcher, F. Κ. Diakonos

Published Mar 2, 2021• 1 citation
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Abstract

We explore two-dimensional scattering off a potential that is invariant with respect to linear symmetry transformations such as rotations, reflections, and/or coordinate exchange. The common ansatz of an incoming wave, in general, does not respect the symmetries of the potential and, therefore, the solution of the corresponding Schr\"odinger equation is not an eigenstate of the operator representing the respective symmetry transform. This renders it difficult to conceptually account for the imprint of the potentials' symmetry on the scattered wave field, unless the limit of an infinite distance $r\ensuremath{\rightarrow}\ensuremath{\infty}$ is taken. In the latter case, the symmetries of the potential are expressed as conditions on the scattering amplitude. In the present work, we employ the recently derived, symmetry-induced, nonlocal and divergence-free currents, which are constructed from the wave field at a point $\mathbf{r}$ and its symmetry-related image $\overline{\mathbf{r}}$, to derive general properties of the scattering solution and the associated scattering matrix. These properties originate from the requirement of a vanishing divergence for the nonlocal currents, which is in one-to-one correspondence with the presence of a symmetry in the scattering potential. In practice, they are expressed as conditions on the coefficients of the wave-field expansion with respect to the angular momentum basis in two dimensions. This, in turn, constrains the form of the related $\mathbf{S}$-matrix eigenvectors. The obtained properties are also valid at finite $\mathbf{r}$ and can be used as a tool to check upon and improve the efficiency of numerical approaches to the quantum scattering problem of interest.

DOI
10.1103/physreva.103.032203

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We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multidelta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and the Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schrodinger equation for an effective nonunitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.
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