Scattering matrix elements and energy spectrum of one-dimensional hybrid PT-symmetric finite systems

TL;DR

This paper analytically characterizes the scattering matrix and energy spectrum of one-dimensional PT-symmetric hybrid systems, revealing conditions for spectral singularities and providing explicit quantization formulas.

Contribution

It introduces a complete analytical framework for describing scattering and spectral properties of finite PT-symmetric systems with gain and loss regions.

Findings

Derived closed-form energy spectrum expression.

Identified conditions for spectral singularities.

Provided a quantization condition for finite lattice systems.

Abstract

In this work, we provide a complete description of the scattering matrix elements and electron energy spectrum in one dimensional PT-symmetric hybrid finite systems, using the characteristic determinant approach. We present an analytical formulation of the problem and obtain a closed-form expression for the energy spectrum of the system, consisting of a region of real potential (passive region) surrounded by regions of gain and loss on the left and right, respectively. It has been shown that under certain conditions and a specific ratio between the real and imaginary parts of the complex potentials, it is possible to find analytical expressions for the spectral singularities at which the scattering matrix elements of the hybrid structure tend to infinity at a specific real energy. Within the framework of the same approach, we present a compact analytical expression for the quantization condition that determines the energy spectrum of a model corresponding to the placement of a rigid lattice within a finite-sized box.