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THE METRIC OPERATORS FOR PSEUDO-HERMITIAN HAMILTONIAN

Part of: General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  23 October 2023

WEN-HUA WANG Open the ORCID record for WEN-HUA WANG [Opens in a new window] ,
ZHENG-LI CHEN Open the ORCID record for ZHENG-LI CHEN [Opens in a new window]  and
WEI LI Open the ORCID record for WEI LI [Opens in a new window]
WEN-HUA WANG
Affiliation:
School of Ethnic Education, Shaanxi Normal University, Xi’an 710062, China; e-mail: wenhua@snnu.edu.cn
ZHENG-LI CHEN*
Affiliation:
School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710119, China
WEI LI
Affiliation:
College of Science, Engineering University of PAP, Xi’an 710078, China; e-mail: 398979528@qq.com
*
e-mail: czl@snnu.edu.cn
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Abstract

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.

Keywords

pseudo-Hermitian Hamiltonianmetric operatorvec mapreal spectrumpositive-definite inner product

MSC classification

Primary: 81Q12: Non-selfadjoint operator theory in quantum theory
Type
Research Article
Information
The ANZIAM Journal , Volume 65 , Issue 3 , July 2023 , pp. 215 - 228
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

Bellman, R., Introduction to matrix analysis (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1987); doi:10.1137/1.9781611971170.fm.Google Scholar
Bender, C. and Boettcher, S., “Real spectra in non-Hermitian Hamiltonians having PT symmetry”, Phys. Rev. Lett. 80 (1998) 5243–5246; doi:10.1103/PhysRevLett.80.5243.CrossRefGoogle Scholar
Bender, C. M., “Making sense of non-Hermitian Hamiltonians”, Rep. Progr. Phys. 70 (2007) 947–1018; 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) 2201–2229; 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) 158–163; 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) 469–478 (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) 205–214; 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) 2814–2816; 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) 3944–3951; doi:10.1063/1.1489072.CrossRefGoogle Scholar
Mostafazadeh, A., “Exact PT-symmetry is equivalent to Hermiticity”, J. Phys. A 36 (2003) 7081–7091; 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) 919–933; 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) 10171–10188; 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) F75–F80; 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) 84–92; 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) 47–101; 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