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The Hamiltonians of classical quantum systems are Hermitian (self-adjoint) operators. The self-adjointness of a Hamiltonian not only ensures that the system follows unitary evolution and preserves probability conservation, but also guarantee that the Hamiltonian has real energy eigenvalues. We call such systems Hermitian quantum systems. However, there exist indeed some physical systems whose Hamiltonians are not Hermitian, for instance,
$ {\mathcal{P}}{\mathcal{T}} $ -symmetry quantum systems. We refer to such systems as non-Hermitian quantum systems. To discuss in depth$ {\mathcal{P}}{\mathcal{T}} $ -symmetry quantum systems, some properties of conjugate linear operators are discussed first in this paper due to the conjugate linearity of the operator$ {\mathcal{P}}{\mathcal{T}}, $ including their matrix represenations, spectral structures, etc. Second, the conjugate linear symmetry and unbroken conjugate linear symmetry are introduced for linear operators, and some equivalent characterizations of unbroken conjugate linear symmetry are obtained in terms of the matrix representations of the operators. As applications,$ {\mathcal{P}}{\mathcal{T}} $ -symmetry and unbroken$ {\mathcal{P}}{\mathcal{T}} $ -symmetry of Hamiltonians are discussed, showing that unbroken$ {\mathcal{P}}{\mathcal{T}} $ -symmetry is not closed under taking tensor-product operation by some specific examples. Moreover, it is also illustrated that the unbroken$ {\mathcal{P}}{\mathcal{T}} $ -symmetry is neither a sufficient condition nor a necessary condition for Hamiltonian to be Hermitian under a new positive definite inner product.-
Keywords:
- conjugate linear operator /
- conjugate linear symmetry /
- unbroken conjugate linear symmetry /
- $ {\mathcal{P}}{\mathcal{T}} $-symmetry /
- unbroken $ {\mathcal{P}}{\mathcal{T}} $-symmetry
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图 1 共轭线性算子
$ {\mathcal{A}} $ 与$ {\mathcal{T}}_0 $ 的复合算子$ {\mathcal{A}}{\mathcal{T}}_0 $ (左)与$ {\mathcal{T}}_0{\mathcal{A}} $ (右)都是线性算子, 且满足关系$ {\mathcal{A}}=({\mathcal{A}}{\mathcal{T}}_0){\mathcal{T}}_0= $ ${\mathcal{T}}_0({\mathcal{T}}_0{\mathcal{A}}) $ Figure 1. Composition operators
$ {\mathcal{A}}{\mathcal{T}}_0 $ (left) and$ {\mathcal{T}}_0{\mathcal{A}} $ (right), composed of conjugate linear operators$ {\mathcal{A}} $ and$ {\mathcal{T}}_0 $ , which are linear operators and satisfy$ {\mathcal{A}}=({\mathcal{A}}{\mathcal{T}}_0){\mathcal{T}}_0={\mathcal{T}}_0({\mathcal{T}}_0{\mathcal{A}}) $ -
[1] Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243
Google Scholar
[2] Bender C M, Berry M V, Mandilara A 2002 J. Phys. A: Math. Theor. 35 L467
Google Scholar
[3] Bender C M, Brody D C, Jones H F 2002 Phys. Rev. Lett. 89 270401
[4] Bender C M, Brandt S F, Chen J H, Wang Q H 2005 Phys. Rev. D 71 025014
Google Scholar
[5] Bender C M 2007 Rep. Prog. Phys. 70 947
Google Scholar
[6] Bender C M, Klevansky S P 2009 Phys. Lett. 373 2670
Google Scholar
[7] Bender C M, Gianfreda M 2013 J. Phys. A: Math. Theor. 46 275306
Google Scholar
[8] Mostafazadeh A 2002 J. Math. Phys. 43 205
[9] Mostafazadeh A 2002 J. Math. Phys. 43 2814
Google Scholar
[10] Mostafazadeh A 2002 J. Math. Phys. 43 3944
Google Scholar
[11] Mostafazadeh A 2007 Phys. Rev. Lett. 99 130502
Google Scholar
[12] Mostafazadeh A 2010 Int. J. Geom. Methods. M 7 1191
Google Scholar
[13] Bender C M, Brody D C, Jones H F, Meister B K 2007 Phys. Rev. Lett. 98 040403
Google Scholar
[14] Zheng C, Hao L, Long G L 2013 Phil. Trans. R. Soc. A 371 20120053
Google Scholar
[15] Rüter E C, Makris G K, Ganainy E R, Christodoulides N D, Segev M, Kip D 2010 Nat. Phys. 6 192
Google Scholar
[16] Bender C M, Mannheim P D 2011 Phys. Rev. D 84 105038
Google Scholar
[17] Kevrekidis P G 2014 Phys. Rev. A 89 010102
Google Scholar
[18] Chen S L, Chen G Y, Chen Y N 2014 Phys. Rev. A 90 054301
Google Scholar
[19] Lee C Y, Hsieh H M, Flammia T S, Lee K R 2014 Phys. Rev. Lett. 112 130404
Google Scholar
[20] Tang J S, Wang Y T, Yu S, He D Y, Xu J S, Liu B H, Chen G, Sun Y N, Sun K, Han Y J, Li C F, Guo G C 2016 Nat. Photonics 10 642
Google Scholar
[21] Mochizuki K, Kim D, Obuse H 2016 Phys. Rev. A 93 062116
Google Scholar
[22] Xiao L, Zhan X, Bian Z H, Wang K K, Zhang X, Wang X P, Li J, Mochizuki K, Kim D, Kawakami N, Yi W, Obuse H, Sanders C B, Xue P 2017 Nat. Phys. 13 1117
Google Scholar
[23] Izaac A J, Wang B J, Abbott C P, Ma S X 2017 Phys. Rev. A 96 032305
Google Scholar
[24] Smith J K, Mathur H 2010 Phys. Rev. A 82 042101
Google Scholar
[25] Ashok D 2011 J. Phys. Conf. Ser. 287 012002
Google Scholar
[26] Longhi S 2011 J. Phys. A: Math. Theor. 44 485302
Google Scholar
[27] Cao H X, Guo Z H, Chen Z L 2013 Commun. Theor. Phys. 60 328
Google Scholar
[28] Guo Z H, Cao H X, Lu L 2014 Sci. China: Phys. Mech. Astron. 57 1835
[29] Deffner S, Saxena A 2015 Phys. Rev. Lett. 114 150601
Google Scholar
[30] Croke S 2015 Phys. Rev. A 91 052113
Google Scholar
[31] Brody D C 2016 J. Phys. A: Math. Theor. 49 10LT03
Google Scholar
[32] Longhi S, Fisica D D 2017 Sci. Bull. 62 869
Google Scholar
[33] Huang M Y, Kumar A, Wu J D 2018 Phys. Lett. A 382 2578
Google Scholar
[34] Huang M Y, Lee K R, Wu J D 2018 J. Phys. A: Math. Theor. 51 414004
Google Scholar
[35] Ramy E G, Konstantinos G M, Mercedeh K, Ziad H M, Stefan R, Demetrios N C 2018 Nat. Phys. 14 11
Google Scholar
[36] Zhu W W, Fang X S, Li D T, Sun Y, Li Y, Jing Y, Chen H 2018 Phys. Rev. Lett. 121 124501
Google Scholar
[37] 黄永峰, 曹怀信, 王文华 2019 数学学报 62 469
Google Scholar
Huang Y F, Cao H X, Wang W H 2019 Acta. Math. Sin. 62 469
Google Scholar
[38] Huang M Y, Lee R K, Zhang L J, Fei S M, Wu J D 2019 Phys. Rev. Lett. 123 080404
Google Scholar
[39] Horn A R, Johnson R C 2013 Matrix Analysis (2nd Ed.) (Cambridge: Cambridge University Press) pp163–187
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