A new solvable complex PT-symmetric potential

TL;DR

This paper introduces a new exactly solvable complex PT-symmetric potential, analyzes its spectral properties, including the existence of real eigenvalues and exceptional points, and explores the relation between eigenstates and PT symmetry.

Contribution

It presents a novel solvable PT-symmetric potential and investigates its spectral characteristics, including eigenvalue behavior and PT symmetry relations.

Findings

Finite real eigenvalues for small parameters

Existence of exceptional points where eigenvalues coalesce

Real eigenvalues as poles of reflection coefficient

Abstract

We propose a new solvable one-dimensional complex PT-symmetric potential as $V(x)= ig~ \mbox{sgn}(x)~ |1-\exp(2|x|/a)|$ and study the spectrum of $H=-d^2/dx^2+V(x)$. For smaller values of $a,g <1$, there is a finite number of real discrete eigenvalues. As $a$ and $g$ increase, there exist exceptional points (EPs), $g_n$ (for fixed values of $a$) causing a scarcity of real discrete eigenvalues, but there exists at least one. We also show these real discrete eigenvalues as poles of reflection coefficient. We find that the energy-eigenstates $\psi_n(x)$ satisfy (1): PT$\psi_n(x)=1 \psi_n(x)$ and (2): PT$\psi_{E_n}(x)=\psi_{E^*_n}(x)$, for real and complex energy eigenvalues, respectively.