Construction of maximally non-Hermitian potentials under unbroken PT-symmetry constraint

TL;DR

This paper investigates the construction of maximally non-Hermitian potentials in discrete Schrödinger equations under unbroken PT-symmetry, analyzing the spectral properties and exceptional points where eigenvalues become degenerate.

Contribution

It introduces a method to construct maximally non-Hermitian PT-symmetric potentials and studies their spectral degeneracies and exceptional points in discrete systems.

Findings

Bound-state energies remain real within the unitarity domain.

Exceptional points mark the boundary of unbroken PT-symmetry.

Complexity increases with the number of grid points.

Abstract

A family of discrete Schr\"{o}dinger equations with imaginary potentials $V(x)$ is studied. Inside the domain ${\cal D}$ of unitarity-compatible values of $V(x)$, the reality of all of the bound-state energies survives up to the ``exceptional-point'' (EP) maximally non-Hermitian spectral-degeneracy boundaries $\partial {\cal D}$. The computer-assisted localization of the EP limits is performed showing that the complexity of the task grows quickly with the number $N$ of grid points $x$.