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Spontaneous PT-Symmetry Breaking for Systems of Noncommutative Euclidean Lie Algebraic Type

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We propose a noncommutative version of the Euclidean Lie algebra E 2. Several types of non-Hermitian Hamiltonian systems expressed in terms of generic combinations of the generators of this algebra are investigated. Using the breakdown of the explicitly constructed Dyson maps as a criterium, we identify the domains in the parameter space in which the Hamiltonians have real energy spectra and determine the exceptional points signifying the crossover into the different types of spontaneously broken PT-symmetric regions with pairs of complex conjugate eigenvalues. We find exceptional points which remain invariant under the deformation as well as exceptional points becoming dependent on the deformation parameter of the algebra.

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  1. Dey, S., Fring, A., Mathanaranjan, T.: Non-Hermitian systems of Euclidean Lie algebraic type with real eigenvalue spectra. Ann. Phys. 346, 28–41 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  2. Turbiner, A.: Lie algebras and linear operators with invariant subspaces. In: Kamran, N., Olver, P.J. (eds.) Lie Algebras, Cohomologies and New Findings in Quantum Mechanics, Contemp. Math. AMS, vol. 160, pp. 263–310 (1994)

  3. Assis, P.E.G., Fring, A.: Non-Hermitian Hamiltonians of Lie algebraic type. J. Phys. A 42, 015203 (23p) (2009)

    ADS  Google Scholar 

  4. Assis, P.E.G.: Metric operators for non-Hermitian quadratic su(2) Hamiltonians. J. Phys. A 44, 265303 (2011)

    Article  ADS  Google Scholar 

  5. Bender, C.M., Kalveks, R.J.: Extending PT symmetry from Heisenberg Algebra to E2 Algebra. Int. J. Theor. Phys. 50, 955–962 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Jones-Smith, K., Kalveks, R.J.: Vector models in PT quantum mechanics. Int. J. Theor. Phys. 52, 2187–2195 (2013)

    Article  MathSciNet  Google Scholar 

  7. Musslimani, Z.H., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Optical solitons in PT periodic potentials. Phys. Rev. Lett. 100, 030402 (2008)

    Article  ADS  Google Scholar 

  8. Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: PT-symmetric optical lattices. Phys. Rev. A 81, 063807(10) (2010)

    Article  ADS  Google Scholar 

  9. Guo, A., Salamo, G.J., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V., Siviloglou, G.A., Christodoulides, D.: Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902(4) (2009)

    ADS  Google Scholar 

  10. Midya, B., Roy, B., Roychoudhury, R.: A note on the PT invariant potential 4c o s 2 x + 4i V 0 s i n2x. Phys. Lett. A 374, 2605–2607 (2010)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Jones, H.: Use of equivalent Hermitian Hamiltonian for PT-symmetric sinusoidal optical lattices. J. Phys. A 44, 345302 (2011)

    Article  MathSciNet  Google Scholar 

  12. Graefe, E., Jones, H.: PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold. Phys. Rev. A 84, 013818(8) (2011)

    Article  ADS  Google Scholar 

  13. Longhi, S., Della Valle, G.: Invisible defects in complex crystals. Ann. Phys. 334, 35–46 (2013)

    Article  MATH  ADS  Google Scholar 

  14. Wigner, E.: Normal form of antiunitary operators. J. Math. Phys. 1, 409–413 (1960)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Bender, C.M., Boettcher, S.: Real spectra in Non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007)

    Article  ADS  Google Scholar 

  17. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)

    Book  MATH  Google Scholar 

  18. Heiss, W.D.: Repulsion of resonance states and exceptional points. Phys. Rev. E 61, 929–932 (2000)

    Article  ADS  Google Scholar 

  19. Rotter, I., Exceptional points and double poles of the S matrix. Phys. Rev. E 67, 026204 (2003)

  20. Günther, U., Rotter, I., Samsonov, B.F.: Projective Hilbert space structures at exceptional points. J. Phys. A: Math. Theoret. 40(30), 8815 (2007)

    Article  MATH  ADS  Google Scholar 

  21. Scholtz, F.G., Geyer, H.B., Hahne, F.: Quasi-Hermitian operators in quantum mechanics and the variational principle. Ann. Phys. 213, 74–101 (1992)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Mostafazadeh, A.: Pseudo-Hermitian representation of quantum mechanics. Int. J. Geom. Methods Mod. Phys. 7, 1191–1306 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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SD is supported by a City University Research Fellowship. TM is funded by an Erasmus Mundus scholarship and thanks City University for kind hospitality.

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Correspondence to Andreas Fring.

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Dey, S., Fring, A. & Mathanaranjan, T. Spontaneous PT-Symmetry Breaking for Systems of Noncommutative Euclidean Lie Algebraic Type. Int J Theor Phys 54, 4027–4033 (2015).

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