Three PT-symmetric Hamiltonians with completely different spectra

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 $g_c$.