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Pseudo-Hermiticity versus \({\mathcal P}{\mathcal T}\)-symmetry. III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries. (English) Zbl 1061.81075

Summary: 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.
For Parts I, II, cf. ibid. 43, No. 1, 205–214 (2002; Zbl 1059.81070), ibid. 43, No. 5, 2814–2816 (2002; Zbl 1060.81022).

MSC:

81U15 Exactly and quasi-solvable systems arising in quantum theory
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