• DOI:10.1063/1.1489072
• Corpus ID: 7096321

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

@article{Mostafazadeh2002PseudoHermiticityVP,
  title={Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries},
  author={Ali Mostafazadeh},
  journal={Journal of Mathematical Physics},
  year={2002},
  volume={43},
  pages={3944-3951}
}

• A. Mostafazadeh
• Published 4 March 2002
• Mathematics
• Journal of Mathematical Physics

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.