Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries

By Ali Mostafazadeh. 2002

709 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/337a9acccea7ce6e3e0d309127cd8ad33980667d

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In Atomic and Molecular Physics, and Optics

In Statistical and Nonlinear Physics

Citations in the corpus

Nonlinear waves in PT -symmetric systems (2016, background): “…is equivalent to the presence of an antilinear symmetry, such as PT symmetry: a diagonalizable operator H with discrete spectrum is (weakly) pseudo-Hermitian if and only if there exists an invertible antilinear operator Ω such that Ω2 = I and [H,Ω] = 0 (Mostafazadeh, 2002c; Solombrino, 2002).”
Biorthogonal quantum mechanics (2013, background): “Nevertheless, associated with such a Hamiltonian is another symmetry Ĉ, whose properties resemble those of a charge operator in quantum field theories, such that 1̂ = ĈP̂ can be used as a metric for Hilbert space [44, 45].”
The ODE/IM correspondence (2001, neutral): “Bender and Boettcher’s observation [23] has sparked a great deal of interest in reality properties in non-Hermitian quantum mechanics; a (small) sample of related work on the reality issue is provided by [52–112].”
Pseudospectra in non-Hermitian quantum mechanics (2014, methodology): “The procedure can be understood by the concept of pseudo-Hermiticity as developed by Mostafazadeh [52, 53, 54]: a CPTsymmetric operator can be transformed into a self-adjoint operator Hsa = H ∗ sa via a similarity transformation Ω, i.e., Hsa = ΩHΩ −1 .”