Three PT-symmetric Hamiltonians with completely different spectra

TL;DR

This paper examines three different PT-symmetric Hamiltonians with a perturbation $igz$, revealing how their spectra vary from entirely real to complex depending on the underlying potential, highlighting diverse spectral behaviors.

Contribution

It introduces three PT-symmetric Hamiltonians with different spectral properties, demonstrating how the spectrum depends on the choice of the central-field potential.

Findings

H with harmonic oscillator potential has a real spectrum for all g

H with hydrogen atom potential has infinitely many complex eigenvalues for all g

H with linear potential exhibits a PT phase transition at a critical g

Abstract

We discuss three Hamiltonians, each with a central-field part $H_{0}$ and a PT-symmetric perturbation $igz$. When $H_{0}$ is the isotropic Harmonic oscillator the spectrum is real for all $g$ because $H$ is isospectral to $H_{0}+g^{2}/2$. When $H_{0}$ is the Hydrogen atom then infinitely many eigenvalues are complex for all $g$. If the potential in $H_{0}$ is linear in the radial variable $r$ then the spectrum of $H$ exhibits real eigenvalues for $0<g<g_{c}$ and a PT phase transition at $g_c$.