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

TL;DR

The paper introduces pseudo-Hermiticity as a unifying framework for real-spectral non-Hermitian Hamiltonians, showing that PT-symmetric systems are encompassed within this broader structure. It formalizes η-pseudo-Hermiticity, derives key spectral and algebraic consequences, and demonstrates how complete biorthonormal eigenbases enforce real spectra or complex-conjugate pairs. It extends the idea to minisuperspace cosmology, where a Wheeler-DeWitt Hamiltonian is η-pseudo-Hermitian, and develops a pseudo-supersymmetric quantum-mechanics formalism to generate isospectral non-Hermitian partners. Finally, it constructs explicit classes of real-spectrum non-Hermitian Hamiltonians and discusses the implications and open questions for pseudo-unitary quantum mechanics.

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.