Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum

Abstract

We give a necessary and sufficient condition for the reality of the spectrum of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal eigenvectors.Recently, we have explored in [1] the basic mathematical structure underlying the spectral properties of P T -symmetric Hamiltonians [2]. In particular, we have shown that these properties are associated with a class of more general (not necessarily Hermitian) Hamiltonians H satisfyingwhere † denotes the adjoint of the corresponding operator and η is a Hermitian invertible linear operator. We have termed such a Hamiltonian 'η-pseudo-Hermitian.' Hermitian and the P T -symmetric Hamiltonians that admit a complete set of biorthonormal eigenvectors constitute subsets of the set of pseudo-Hermitian Hamiltonians. For a P T -symmetricHamiltonian, the exactness of P T -symmetry ensures the reality of the energy spectrum. The purpose of this article is to provide a complete characterization of the Hamiltonians that *

Smeared quantum lattices exhibiting 𝒫𝒯‐symmetry with positive 𝒫

“…(10), the closed-form definition of the biorthogonal basis. Subsequently, such knowledge of the basis converted formula (11) into an explicit definition of all of the eligible metrics and physical Hilbert spaces H (QT ) .…”

“…where Θ (QT ) is given by formula (11). Thus, assuming that we know the (curly bra-ket denoted) eigenstates and spectral representation of…”

“…This is a useful convention because now, the complete set of the eligible PTQT (or, rather, QTQT) metrics may be defined by Mostafazadeh's expression [11,12] …”

“…= I, and at which our two alternative Hilbert spaces, viz., spaces K Q and H (QT ) would coincide. Vice versa, a nontrivial QTQT operator C (QT ) = Q −1 Θ (QT ) = I is obtained whenever we choose, in (11), any other N−plet of parameters,…”

The Hilbert-Space Structure of Non-Hermitian Theories with Real Spectra

“…• the notion of pseudo-Hermiticity, advocated in the work of Mostafazadeh [4] (see also [5] for an early discussion of this concept). Here a metric operator η is used to define a modified scalar product,…”