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Pseudo-Hermiticity versus \(PT\) symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian. (English) Zbl 1059.81070

Summary: 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 \(pseudosupersymmetric\) \(quantum\) \(mechanics\), 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.

MSC:

81Q60 Supersymmetry and quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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