Non-Hermitian Hamiltonians and similarity transformations

TL;DR

This paper demonstrates how similarity transformations can be used to construct non-Hermitian operators with real spectra, linking them to pseudo-Hermiticity and analyzing a three-parameter non-Hermitian oscillator.

Contribution

It introduces a unified approach using similarity transformations to generate non-Hermitian Hamiltonians with real spectra and clarifies their connection to pseudo-Hermiticity.

Findings

Similarity transformations produce non-Hermitian operators with real spectra.

The approach generalizes previous gauge-like transformation results.

Analysis of eigenvalues of a three-parameter non-Hermitian oscillator.

Abstract

We show that similarity (or equivalent) transformations enable one to construct non-Hermitian operators with real spectrum. In this way we can also prove and generalize the results obtained by other authors by means of a gauge-like transformation and its generalization. Such similarity transformations also reveal the connection with pseudo-Hermiticity in a simple and straightforward way. In addition to it we consider the positive and negative eigenvalues of a three-parameter non-Hermitian oscillator.