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

2001/07/31 by Ali Mostafazadeh · 3 citations
Physics and Astronomy · Mathematics · #math-ph #hep-th #math.MP #quant-ph

paper · pdf · doi:10.1063/1.1418246

published as J. Math. Phys. 43, 205-214 (2002) · Revised version, to appear in J. Math. Phys

arxiv created 2001/10/10 · arxiv updated 2016/09/07

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.

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