Dynamical Theory of Scattering, Exact Unidirectional Invisibility, and Truncated $\mathfrak{z}{e}^{-2i{k}_{0}x}$ potential

Abstract

The dynamical formulation of time-independent scattering theory that is developed in [Ann. Phys. (NY) 341, 77-85 (2014)] offers simple formulas for the reflection and transmission amplitudes of finite-range potentials in terms of the solution of an initial-value differential equation. We prove a theorem that simplifies the application of this result and use it to give a complete characterization of the invisible configurations of the truncated $\mathfrak{z}{e}^{-2i{k}_{0}x}$ potential to a closed interval, $[0,L]$, with $k_0$ being a positive integer multiple of $\pi /L$. This reveals a large class of exact unidirectionally and bidirectionally invisible configurations of this potential. The former arise for particular values of $\mathfrak{z}$ that are given by certain zeros of Bessel functions. The latter occur when the wavenumber $k$ is an integer multiple of $\pi /L$ but not of $k_0$. We discuss the optical realizations of these configurations and explore spectral singularities of this potential.

16 pages, 1 figure, 1 table

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• The transfer matrix: a geometrical perspective
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• Transfer Matrix Formulation of Scattering Theory in Two and Three Dimensions
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Cited by in corpus (6)

• Symmetry-Protected Scattering in Non-Hermitian Linear Systems
• Quasi-Exactly Solvable Scattering Problems, Exactness of the Born Approximation, and Broadband Unidirectional Invisibility in Two Dimensions
• Non-Hermitian invisibility in tight-binding lattices
• Unidirectional Reflection and Invisibility in Nonlinear Media with an Incoherent Nonlinearity
• Scattering Theory and $\mathcal{P}\mathcal{T}$-Symmetry
• Scattering of TE and TM waves and quantum dynamics generated by non-Hermitian Hamiltonians