PT-symmetric potentials with imaginary asymptotic saturation

TL;DR

This paper investigates PT-symmetric potentials with imaginary asymptotic saturation, revealing they lack scattering states and spectral singularities, and explores their unique spectral properties including real and complex eigenvalues.

Contribution

It introduces and analyzes a class of PT-symmetric potentials with imaginary asymptotic saturation, detailing their spectral characteristics and eigenstate behaviors, which were not previously documented.

Findings

No scattering states or spectral singularities in these potentials.

Existence of real discrete spectra with and without complex conjugate pairs.

Eigenstates exhibit symmetry and asymmetry properties, with oscillatory behaviors.

Abstract

We point out that PT-symmetric potentials $V_{PT}(x)$ having imaginary asymptotic saturation: $V_{PT}(x=\pm \infty) =\pm i V_1, V_1 \in \Re$ are devoid of scattering states and spectral singularity. We show the existence of real (positive and negative) discrete spectrum both with and without complex conjugate pair(s) of eigenvalues (CCPEs). If the states are arranged in the ascending order or real part of discrete eigenvalues, the initial states have few nodes but latter ones oscillate fast. Both real and imaginary parts of $\psi(x)$ vanish asymptotically, $|\psi(x)|$ for the CCPEs are asymmetric and for real energies these are symmetric about origin. For CCPEs $E_{\pm}$ the eigenstates $\psi_{\pm}$ follow an interesting property that $|\psi_+(x)|= N |\psi_-(-x)|, N \in \Re^+$. We remark that, the fast oscillating real discrete energy states discussed are likely to be confused with: reflectionless states, one dimensional version of von Neumann states of Hermitian and spectral singularity state of complex PT-symmetric potentials.