Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-20T04:40:25.763Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  23 October 2023

School of Ethnic Education, Shaanxi Normal University, Xi’an 710062, China; e-mail:
School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710119, China
College of Science, Engineering University of PAP, Xi’an 710078, China; e-mail:


The Hamiltonian of a conventional quantum system is Hermitian, which ensures real spectra of the Hamiltonian and unitary evolution of the system. However, real spectra are just the necessary conditions for a Hamiltonian to be Hermitian. In this paper, we discuss the metric operators for pseudo-Hermitian Hamiltonian which is similar to its adjoint. We first present some properties of the metric operators for pseudo-Hermitian Hamiltonians and obtain a sufficient and necessary condition for an invertible operator to be a metric operator for a given pseudo-Hermitian Hamiltonian. When the pseudo-Hermitian Hamiltonian has real spectra, we provide a new method such that any given metric operator can be transformed into the same positive-definite one and the new inner product with respect to the positive-definite metric operator is well defined. Finally, we illustrate the results obtained with an example.

Research Article
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Bellman, R., Introduction to matrix analysis (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1987); doi:10.1137/ Scholar
Bender, C. and Boettcher, S., “Real spectra in non-Hermitian Hamiltonians having PT symmetry”, Phys. Rev. Lett. 80 (1998) 52435246; doi:10.1103/PhysRevLett.80.5243.CrossRefGoogle Scholar
Bender, C. M., “Making sense of non-Hermitian Hamiltonians”, Rep. Progr. Phys. 70 (2007) 9471018; doi:10.1088/0034-4885/70/6/R03.CrossRefGoogle Scholar
Bender, C. M., Boettcher, S. and Meisinger, P. N., “PT-symmetric quantum mechanics”, J. Math. Phys. 40 (1999) 22012229; doi:10.1063/1.532860.CrossRefGoogle Scholar
Feinberg, J. and Riser, R., “Pseudo-Hermitian random matrix theory: A review”, J. Phys.: Conf. Ser. 2038 (2021) Article ID 012009; doi:10.1088/1742-6596/2038/1/012009.Google Scholar
Feinberg, J. and Znojil, M., “Which metrics are consistent with a given pseudo-Hermitian matrix”, J. Math. Phys. 63 (2022) Article ID 013505; doi:10.1063/5.0079385.CrossRefGoogle Scholar
Fring, A. and Frith, T., “Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians”, Phys. Lett. A 383 (2019) 158163; doi:10.1016/j.physleta.2018.10.043.CrossRefGoogle Scholar
Galeano, D. A., Zhang, X. X. and Mahecha, J., “Topological circuit of a versatile non-Hermitian quantum system”, Sci. China-Phys. Mech. Astron. 65 (2022) Article ID 217211; doi:10.1007/s11433-021-1783-3.CrossRefGoogle Scholar
Gopalakrishnan, S. and Gullans, M. J., “Entanglement and purification transitions in non-Hermitian quantum mechanics”, Phys. Rev. Lett. 126 (2021) Article ID 170503; doi:10.1103/PhysRevLett.126.170503.CrossRefGoogle ScholarPubMed
Huang, Y. F., Cao, H. X. and Wang, W. H., “Unitary evolution and adiabatic theorem of pseudo self-adjoint quantum systems”, Acta Math. Sin. 62 (2019) 469478 (in Chinese); doi:10.12386/A2019sxxb0044.Google Scholar
Kawabata, K. and Sato, M., “Real spectra in non-Hermitian topological insulators”, Phys. Rev. Res. 2 (2020) Article ID 033391; doi:10.1103/PhysRevResearch.2.033391.CrossRefGoogle Scholar
Mostafazadeh, A., “Pseudo-Hermiticity versus PT-symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian”, J. Math. Phys. 43 (2002) 205214; doi:10.1063/1.1418246.CrossRefGoogle Scholar
Mostafazadeh, A., “Pseudo-Hermiticity versus PT-symmetry II: A complete characterization of non-Hermitian Hamiltonians with a real spectrum”, J. Math. Phys. 43 (2002) 28142816; doi:10.1063/1.1461427.CrossRefGoogle Scholar
Mostafazadeh, A., “Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries”, J. Math. Phys. 43 (2002) 39443951; doi:10.1063/1.1489072.CrossRefGoogle Scholar
Mostafazadeh, A., “Exact PT-symmetry is equivalent to Hermiticity”, J. Phys. A 36 (2003) 70817091; doi:10.1088/0305-4470/36/25/312.CrossRefGoogle Scholar
Mostafazadeh, A., “Krein-space formulation of PT-symmetry, CPT-inner products, and pseudo-Hermiticity”, Czech. J. Phys. 56 (2006) 919933; doi:10.1007/s10582-006-0388-8.CrossRefGoogle Scholar
Mostafazadeh, A., “Metric operator in pseudo-Hermitian quantum mechanics and the imaginary cubic potential”, J. Phys. A 39 (2006) 1017110188; doi:10.1088/0305-4470/39/32/S18.CrossRefGoogle Scholar
Musumbu, D. P., Geyer, H. B. and Heiss, W. D., “Choice of a metric for the non-Hermitian oscillator”, J. Phys. A 40 (2007) F75F80; doi:10.1088/1751-8113/40/2/F03.CrossRefGoogle Scholar
Raimundo, K., Baldiotti, M. C., Fresneda, R. and Molina, C., “Classical-quantum correspondence for two-level pseudo-Hermitian systems”, Phys. Rev. A 103 (2021) Article ID 022201; doi:10.1103/PhysRevA.103.022201.CrossRefGoogle Scholar
Scholtz, F. G. and Geyer, H. B., “Operator equations and Moyal products-metrics in quasi-Hermitian quantum mechanics”, Phys. Lett. B 634 (2006) 8492; doi:10.1016/j.physletb.2006.01.022.CrossRefGoogle Scholar
Scholtz, F. G., Geyer, H. B. and Hahne, F. J. W., “Quasi-Hermitian operators in quantum mechanics and the variational principle”, Ann. Phys. 213 (1992) 47101; doi:10.1016/0003-4916(92)90284-S.CrossRefGoogle Scholar
Zhan, X., Wang, K., Xiao, L., Bian, Z., Zhang, Y., Sanders, B. C., Zhang, C. and Xue, P., “Experimental quantum cloning in a pseudo-unitary system”, Phys. Rev. A 101 (2020) Article ID 010302(R); doi:10.1103/PhysRevA.101.010302.CrossRefGoogle Scholar
Zhang, R. L., Qin, H. and Xiao, J. Y., “PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability”, J. Math. Phys. 61 (2020) Article ID 012101; doi:10.1063/1.5117211.CrossRefGoogle Scholar