Skip to main content
Log in

Scattering Theory for 0-Perturbed \( \mathcal{P}\mathcal{T} \) -Symmetric Operators

  • Published:
Ukrainian Mathematical Journal Aims and scope

The aim of the present work is to develop the scattering theory for 0-perturbed \( \mathcal{P}\mathcal{T} \) -symmetric operators by using the Lax–Phillips method. The presence of a stable \( \mathcal{C} \) -symmetry leading to the property of selfadjointness (with proper choice of the inner product) for these \( \mathcal{P}\mathcal{T} \) -symmetric operators is described in terms of the corresponding S -matrix (scattering matrix).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. Caliceti, F. Cannata, and S. Graffi, “\( \mathcal{P}\mathcal{T} \) -symmetric Schrödinger operators, reality of the perturbed eigenvalues,” SIGMA, 6, 9–17 (2010).

    MathSciNet  Google Scholar 

  2. U. Günther and S. Kuzhel, “\( \mathcal{P}\mathcal{T} \) -symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras,” J. Phys. A: Math. Theor., 43, No. 39, 392002–392011 (2010).

    Article  Google Scholar 

  3. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rept. Progr. Phys., 70, No. 6, 947–1018 (2007).

    Article  Google Scholar 

  4. A. Mostafazadeh, “Pseudo-Hermitian representation of quantum mechanics,” Int. J. Geom. Meth. Mod. Phys., 7, 1191–1306 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  5. Ahmed Zafar, C. M. Bender, and M. V. Berry, “Reflectionless potentials and \( \mathcal{P}\mathcal{T} \) symmetry,” J. Phys. A, 38, L627–L630 (2005).

    Article  MATH  Google Scholar 

  6. F. Cannata, J.-P. Dedonder, and A.Ventura, “Scattering in \( \mathcal{P}\mathcal{T} \) -symmetric quantum mechanics,” Ann. Phys., 322, 397–433 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  7. H. F. Jones, “Scattering from localized non-Hermitian potentials,” Phys. Rev. D, 76, 125003–125008 (2007).

    Article  Google Scholar 

  8. H. Hernandez-Coronado, D. Krejčiřík, and P. Siegl, “Perfect transmission scattering as a \( \mathcal{P}\mathcal{T} \) -symmetric spectral problem,” Phys. Lett. A, 375, 2149–2152 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Mostafazadeh, “Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies,” Phys. Rev. Lett., 102, 220402–220407 (2009).

    Article  Google Scholar 

  10. M. Znojil, “Scattering theory with localized non-Hermiticities,” Phys. Rev. D, 78, 025026–025036 (2008).

    Article  MathSciNet  Google Scholar 

  11. S. Albeverio and S. Kuzhel, “On elements of the Lax–Phillips scattering scheme for \( \mathcal{P}\mathcal{T} \) -symmetric operators,” J. Phys. A: Math. Theor., 45, 1–21 (2012).

    Article  MathSciNet  Google Scholar 

  12. P. A. Cojuhari and S. Kuzhel, “Lax–Phillips scattering theory for \( \mathcal{P}\mathcal{T} \) -symmetric ρ-perturbed operators,” J. Math. Phys., 53, 073514–073531 (2012).

    Article  MathSciNet  Google Scholar 

  13. P. Lax and R. Phillips, Scattering Theory, Academic Press, New York (1989).

    MATH  Google Scholar 

  14. S. Kuzhel, “On the inverse problem in the Lax–Phillips scattering theory method for a class of operator-differential equations,” St. Petersburg Math. J., 13, 41–56 (2002).

    MATH  MathSciNet  Google Scholar 

  15. S. Kuzhel and U. Moskalyova, “The Lax–Phillips scattering approach and singular perturbations of Schrödinger operator homogeneous with respect to scaling transformations,” J. Math. Kyoto Univ., 45, No. 2, 265–286 (2005).

    MATH  MathSciNet  Google Scholar 

  16. N. Dunford and J. T. Schwartz, Linear Operators. Vol. II. Spectral Theory. Self-Adjoint Operators in Hilbert Spaces, Interscience, New York (1963).

  17. V. I. Gorbachuk and M. L. Gorbachuk, Boundary-Value Problems for Differential-Operator Equations [in Russian], Naukova Dumka, Kiev (1984).

    Google Scholar 

  18. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Spaces [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  19. V. A. Derkach and M. M. Malamud, “Characteristic functions of almost solvable extensions of Hermitian operators,” Ukr. Mat. Zh., 44, No. 4, 435–459 ((1992); English translation: Ukr. Math. J., 44, No. 4, 379–401 (1992).

    Article  MathSciNet  Google Scholar 

  20. S. Hassi and S. Kuzhel, “On J-self-adjoint operators with stable \( \mathcal{C} \) -symmetries,” Proc. Roy. Soc. Edinburgh: Sect. A. Math., 143 (2013).

  21. S. Kuzhel and C. Trunk, “On a class of J-self-adjoint-operators with empty resolvent set,” J. Math. Anal. Appl., 379, Issue 1, 272–289 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  22. A. I. Hrod, “On the theory of \( \mathcal{P}\mathcal{T} \) -symmetric operators,” Cherniv. Nauk. Visn., 1, No. 4, 36–42 (2011).

    Google Scholar 

  23. O. M. Patsyuck, “On stable C-symmetries for a class of PT -symmetric operators,” Meth. Funct. Anal. Top., 19, No. 1 (2013).

    Google Scholar 

  24. S. Albeverio and S. Kuzhel, “One-dimensional Schrödinger operators with \( \mathcal{P} \) -symmetric zero-range potentials,” J. Phys. A, 38, 4975–4988 (2005).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 8, pp. 1059–1079, August, 2013.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hrod, A.I., Kuzhel’, S.O. Scattering Theory for 0-Perturbed \( \mathcal{P}\mathcal{T} \) -Symmetric Operators. Ukr Math J 65, 1180–1202 (2014). https://doi.org/10.1007/s11253-014-0851-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-014-0851-9

Keywords

Navigation