Comment on: `Numerical estimates of the spectrum for anharmonic PT symmetric potentials' [Phys. Scr. \textbf{85} (2012) 065005]

TL;DR

This paper critiques a previous study on anharmonic PT-symmetric potentials, highlighting issues with eigenvalue convergence, misinterpretation of real eigenvalues, and misreading of foundational arguments in the field.

Contribution

It clarifies the correct interpretation of eigenvalues in PT-symmetric Hamiltonians and addresses methodological errors in the criticized work.

Findings

Eigenvalues of truncated Hamiltonians do not always converge with increasing matrix size.

Real positive eigenvalues are often missed when focusing on complex eigenvalues.

Misinterpretation of Bender's argument led to incorrect conclusions about eigenvalue spectra.

Abstract

We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operator and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-$x$ plane (Rep. Prog. Phys. {\bf 70} (2007) 947).