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arXiv:math-ph/0110016 · Journal of Mathematical Physics

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

Ali Mostafazadeh

Published May 1, 2002• 828 citations
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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.
DOI
10.1063/1.1461427

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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.
We show that a diagonalizable (non-Hermitian) Hamiltonian H is pseudo-Hermitian if and only if it has an antilinear symmetry, i.e., a symmetry generated by an invertible antilinear operator. This implies that the eigenvalues of H are real or come in complex conjugate pairs if and only if H possesses such a symmetry. In particular, the reality of the spectrum of H implies the presence of an antilinear symmetry. We further show that the spectrum of H is real if and only if there is a positive-definite inner-product on the Hilbert space with respect to which H is Hermitian or alternatively there is a pseudo-canonical transformation of the Hilbert space that maps H into a Hermitian operator.
We notice that non-linear oscillator Hamiltonian having the character of non-Hermitian (H≠H+)(H \ne H^{+})(H=H+), non-PT Symmetry ([H,PT]≠0)([H,PT] \ne 0)([H,PT]=0) and non-Pseudohermiticity (ηHη−1≠H+)(\eta H \eta^{-1}\ne H^{+})(ηHη−1=H+) can yield real eigenvalues.