Scattering Theory and $\mathcal{P}\mathcal{T}$-Symmetry

TL;DR

This paper presents a comprehensive framework for one-dimensional scattering theory that encompasses various symmetries, including $ ext{P}$, $ ext{T}$, and $ ext{PT}$, and explores their implications on scattering properties and spectral singularities.

Contribution

It introduces a unified approach to scattering theory that covers real, complex, energy-dependent, and nonlocal potentials, and derives generalized unitarity relations for symmetric systems.

Findings

Derived conditions for reciprocity and spectral singularities.

Established generalized unitarity relations for $ ext{PT}$-symmetric systems.

Analyzed the effects of symmetry breaking on scattering properties.

Abstract

We outline a global approach to scattering theory in one dimension that allows for the description of a large class of scattering systems and their $\mathcal{P}$-, $\mathcal{T}$-, and $\mathcal{P}\mathcal{T}$-symmetries. In particular, we review various relevant concepts such as Jost solutions, transfer and scattering matrices, reciprocity principle, unidirectional reflection and invisibility, and spectral singularities. We discuss in some detail the mathematical conditions that imply or forbid reciprocal transmission, reciprocal reflection, and the presence of spectral singularities and their time-reversal. We also derive generalized unitarity relations for time-reversal-invariant and $\mathcal{P}\mathcal{T}$-symmetric scattering systems, and explore the consequences of breaking them. The results reported here apply to the scattering systems defined by a real or complex local potential as well as those determined by energy-dependent potentials, nonlocal potentials, and general point interactions.