Three types of discrete energy eigenvalues in complex PT-symmetric scattering potentials

TL;DR

This paper classifies three types of discrete energy eigenvalues in complex PT-symmetric scattering potentials, revealing their conditions, relationships, and the existence of spectral singularities alongside conjugate pair eigenvalues.

Contribution

It introduces a classification of discrete eigenvalues in CPTSSPs, including real, complex conjugate, and spectral singularities, supported by analytical and numerical models.

Findings

Spectral singularities occur at specific potential parameters.

Complex conjugate pair eigenvalues can coexist with spectral singularities.

At most one spectral singularity exists for a fixed potential parameter.

Abstract

For complex PT-symmetric scattering potentials (CPTSSPs) $V(x)= V_1 f_{even}(x) + iV_2 f_{odd}(x), f_{even}(\pm \infty) = 0 = f_{odd}(\pm \infty), V_1,V_2 \in \Re $, we show that complex $k$-poles of transmission amplitude $t(k)$ or zeros of $1/t(k)$ of the type $\pm k_1+ik_2, k_2\ge 0$ are physical which yield three types of discrete energy eigenvalues of the potential. These discrete energies are real negative, complex conjugate pair(s) of eigenvalues (CCPEs: ${\cal E}_n \pm i \gamma_n$) and real positive energy called spectral singularity (SS) at $E=E_*$ where the transmission and reflection co-efficient of $V(x)$ become infinite for a special critical value of $V_2=V_*$. Based on four analytically solvable and other numerically solved models, we conjecture that a parametrically fixed CPTSSP has at most one SS. When $V_1$ is fixed and $V_2$ is varied there may exist Kato's exceptional point(s) $(V_{EP})$ and critical values $V_{*m}, m=0,1,2,..$, so when $V_2$ crosses one of these special values a new CCPE is created. When $V_2$ equals a critical value $V_{*m}$ there exist one SS at $E=E_*$ along with $m$ or more number of CCPEs. Hence, this single positive energy $E_*$ is the upper (or rough upper) bound to the CCPEs: ${\cal E}_l \lessapprox E_*$, here ${\cal E}_l$ corresponds to the last of CCPEs. If $V(x)$ has Kato's exceptional points (EPs: $V_{EP1}<V_{EP2}<V_{EP3}<...<V_{EPl}$), the smallest of critical values $V_{*m}$ is always larger than $V_{EPl}$. Hence, in a CPTSSP, real discrete eigenvalue(s) and the SS are mutually exclusive whereas CCPEs and the SS can co-exist .