## Pseudo-Hermiticity and generalized $$PT$$- and $$CPT$$-symmetries.(English)Zbl 1061.81076

Summary: We study certain linear and antilinear symmetry generators and involution operators associated with pseudo-Hermitian Hamiltonians and show that the theory of pseudo-Hermitian operators provides a simple explanation for the recent results of C. M. Bender, D. C. Brody and H. F. Jones [Phys. Rev. Lett. 89, No. 27, 270401 (2002), see also quant-ph/0208076] on the $$CPT$$-symmetry of a class of $$PT$$-symmetric non-Hermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudo-Hermitian Hamiltonians $$H$$ with a discrete spectrum. In particular, we introduce generalized parity $$(P)$$, time-reversal $$(T)$$, and charge-conjugation $$(C)$$ operators and establish the $$PT$$- and $$CPT$$-invariance of $$H$$.

### MSC:

 81U15 Exactly and quasi-solvable systems arising in quantum theory 47N50 Applications of operator theory in the physical sciences
Full Text:

### References:

 [1] C. M. Bender, D. C. Brody, and H. F. Jones, ”Complex Extension of Quantum Mechanics,” arXiv: quant-ph/0208076. · Zbl 1267.81234 [2] DOI: 10.1063/1.532860 · Zbl 1057.81512 [3] DOI: 10.1063/1.532860 · Zbl 1057.81512 [4] DOI: 10.1063/1.1418246 · Zbl 1059.81070 [5] DOI: 10.1063/1.1461427 · Zbl 1060.81022 [6] DOI: 10.1063/1.1489072 · Zbl 1061.81075 [7] Mostafazadeh A., Nucl. Phys. B 640 pp 419– (2002) · Zbl 0997.81031 [8] Mostafazadeh A., J. Math. Phys. 43 pp 6343– (2002) · Zbl 1060.81023 [9] Scholtz F. G., Ann. Phys. 213 pp 74– (1992) · Zbl 0749.47041 [10] Mostafazadeh A., Class. Quantum Grav. 20 pp 155– (2003) · Zbl 1039.83005 [11] A. Mostafazadeh, ”On a Factorization of Symmertic Matrices and Antilinear Symmerties,” ArXiv: math-ph/0203023. [12] Mostafazadeh A., Mod. Phys. Lett. A 17 pp 1973– (2002) · Zbl 1083.81514
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.