International Journal of Theoretical Physics

, Volume 54, Issue 11, pp 3986–3990 | Cite as

Singular Mapping for a PT-Symmetric Sinusoidal Optical Lattice at the Symmetry-Breaking Threshold

  • H. F. Jones


A popular PT-symmetric optical potential (variation of the refractive index) that supports a variety of interesting and unusual phenomena is the imaginary exponential, the limiting case of the potential \(V_{0}[\cos (2\uppi x/a)+i\uplambda \sin (2\uppi x/a)]\) as λ → 1, the symmetry-breaking point. For λ<1, when the spectrum is entirely real, there is a well-known mapping by a similarity transformation to an equivalent Hermitian potential. However, as λ→1, the spectrum, while remaining real, contains Jordan blocks in which eigenvalues and the corresponding eigenfunctions coincide. In this limit the similarity transformation becomes singular. Nonetheless, we show that the mapping from the original potential to its Hermitian counterpart can still be implemented; however, the inverse mapping breaks down. We also illuminate the role of Jordan associated functions in the original problem, showing that they map onto eigenfunctions in the associated Hermitian problem.


Similarity Transformation Inverse Transformation Negative Integer Jordan Block Paraxial Approximation 
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  1. 1.
    Bender, C.M., Boettcher, S.: Phys. Rev. Lett. 80, 5243 (1998)MATHMathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Dorey, P., Dunning, C., Tateo, R.: J. Phys. A 34, 5679 (2001)MATHMathSciNetCrossRefADSGoogle Scholar
  3. 3.
    Mostafazadeh, A.: J. Math. Phys. 43, 205 (2002)MATHMathSciNetCrossRefADSGoogle Scholar
  4. 4.
    Bender, C.M., Kuzhel, S.: J. Phys. A 45, 444005 (2012)MathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Siegl, P., Krejčiřík, D.: Phys. Rev. D 86, 121702 (2012)Google Scholar
  6. 6.
    El-Ganainy, R., et al.: Opt. Lett. 32, 2632 (2007)CrossRefADSGoogle Scholar
  7. 7.
    Musslimani, Z., et al.: Phy. Rev. Lett. 100, 030402 (2008)CrossRefADSGoogle Scholar
  8. 8.
    Makris, K., et al.: Phys. Rev. Lett. 100, 103904. (2008); Phys. Rev. A 81, 063807 (2010)Google Scholar
  9. 9.
    Longhi, S.: Phys. Rev. A 81, 022102 (2010)CrossRefGoogle Scholar
  10. 10.
    Graefe, E-M., Jones, H.F.: Phys. Rev. A 84, 013818 (2011)CrossRefADSGoogle Scholar
  11. 11.
    Lin, Z., et al.: Phys. Rev. Lett. 106, 213901 (2011)CrossRefADSGoogle Scholar
  12. 12.
    Longhi, S.: J. Phys. A 44, 485302 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jones, H.F.: J. Phys. A 45, 135306 (2012)MathSciNetCrossRefADSGoogle Scholar
  14. 14.
    Midya, N., Roy, B., Choudhury, R.: Phys. Lett. A 374, 2605 (2010)MATHMathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Jones, H.F.: J. Phys. A 44, 345302 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Tables. Dover, New York (1970)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Physics DepartmentImperial CollegeLondonUK

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