Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian

TL;DR

This paper establishes that pseudo-Hermiticity is a necessary condition for non-Hermitian Hamiltonians to have real spectra, unifying PT-symmetric Hamiltonians within this framework and exploring their properties and examples.

Contribution

It introduces pseudo-Hermiticity as a fundamental concept, linking it to real spectra and PT symmetry, and develops pseudo-supersymmetric quantum mechanics.

Findings

All Hamiltonians with real spectra are pseudo-Hermitian.

PT-symmetric Hamiltonians are a subset of pseudo-Hermitian Hamiltonians.

Concrete examples demonstrate the theory's applicability.

Abstract

We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.