Bound states of PT-symmetric separable potentials

TL;DR

This paper investigates nonlocal PT-symmetric separable potentials, demonstrating regions where the Hamiltonian maintains unbroken PT symmetry with real bound-state energies, and calculating the critical coupling strengths that define these regions.

Contribution

It introduces and analyzes nonlocal PT-symmetric separable potentials, extending the study beyond local potentials and identifying conditions for real bound-state energies.

Findings

Existence of parametric regions with unbroken PT symmetry

Calculation of critical coupling strengths for PT symmetry breaking

Demonstration with two specific models

Abstract

All of the PT-symmetric potentials that have been studied so far have been local. In this paper nonlocal PT-symmetric separable potentials of the form $V(x,y)=i\epsilon[U(x)U(y)-U(-x)U(-y)]$, where $U(x)$ is real, are examined. Two specific models are examined. In each case it is shown that there is a parametric region of the coupling strength $\epsilon$ for which the PT symmetry of the Hamiltonian is unbroken and the bound-state energies are real. The critical values of $\epsilon$ that bound this region are calculated.