Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries

TL;DR

This paper establishes that a diagonalizable non-Hermitian Hamiltonian is pseudo-Hermitian if and only if it has an antilinear symmetry, linking spectral reality to such symmetries and inner-product structures.

Contribution

It proves the equivalence between pseudo-Hermiticity and the presence of antilinear symmetries for diagonalizable Hamiltonians, clarifying conditions for real spectra.

Findings

Pseudo-Hermiticity is equivalent to antilinear symmetry for diagonalizable Hamiltonians.

Real spectra imply the existence of a positive-definite inner product making the Hamiltonian Hermitian.

The spectrum's reality is characterized by pseudo-canonical transformations mapping H to a Hermitian operator.

Abstract

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.