International Journal of Theoretical Physics

, Volume 53, Issue 8, pp 2549–2557 | Cite as

The Large −g Observability of the Low-Lying Energies in the Strongly Singular Potentials V (x) = x2 + g2/x6 after their 𝓟T −symmetric Regularization

Article

Abstract

The elementary quadratic plus inverse sextic interaction V (x) = x2 + g2/x6 containing a strongly singular repulsive core in the origin is made regular by a complex shift of coordinate x = s −iε. The shift ε > 0 is fixed while the value of s is kept real and potentially observable, s ∈ (−∞, ∞). The low-lying energies of bound states are found in closed form for the large couplings g ≫ 1. Within the asymptotically vanishing 𝒪(g−1/4) error bars these energies are real so that the time-evolution of the system may be expected unitary in an ad hoc physical Hilbert space.

Keywords

Quantum evolution Triple-Hilbert-space picture Strongly singular forces Regularization by complexification Strong-coupling dynamical regime Unitarity 

References

  1. 1.
    Scholtz, F.G., Geyer, H.B., Hahne, F.J.W.: Ann. Phys. (NY) 213, 74 (1992)ADSCrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bender, C.M.: Rep. Prog. Phys. 70, 947 (2007)ADSCrossRefGoogle Scholar
  3. 3.
    Mostafazadeh, A.: Int. J. Geom. Meth. Mod. Phys. 7, 1191 (2010)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Znojil, M.: SIGMA 5, 001 (2009). arXiv:0901.0700 MathSciNetGoogle Scholar
  5. 5.
    Bender, C.M., Boettcher, S.: Phys. Rev. Lett. 24, 5243 (1988)MathSciNetGoogle Scholar
  6. 6.
    Bender, C.M., Boettcher, S., Meisinger, P.N.: J. Math. Phys. 40, 2201 (1999)ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Buslaev, V., Grecchi, V.: J. Phys. A: Math. Gen. 26, 5541 (1993)ADSCrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Znojil, M.: J. Phys. A: Math. Gen. 33, 4561 (2000)ADSCrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Znojil, M.: J. Phys. A: Math. Gen. 37, 10209 (2004)ADSCrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Harrell, E.M.: Ann. Phys. 105, 379 (1977)ADSCrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Fernández, F.M.: Introduction to Perturbation Theory in Quantum Mechanics. CRC Press, Boca Raton (2001)MATHGoogle Scholar
  12. 12.
    Znojil, M.: Phys. Lett. A 158, 436 (1991)ADSCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bjerrum-Bohr, N.E.J.: J. Math. Phys. 41, 2515 (2000)ADSCrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Bazhanov, V.V., Lukyanov, S.L., Zamolodchikov, A.B.: Adv. Theor. Math. Phys. 7, 711 (2004)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Kratzer, A.: Z. Phys. 3, 289 (1920)ADSCrossRefGoogle Scholar
  16. 16.
    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics, ch. V, par. 35. Pergamon, London (1960)Google Scholar
  17. 17.
    Sibuya, Y.: Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient. Elsevier (1975)Google Scholar
  18. 18.
    Hille, E.: Ordinary Differential Equations in the Complex Domain. Wiley (1976)Google Scholar
  19. 19.
    Dorey, P., Dunning, C., Tateo, R.: J. Phys. A: Math. Theor. 40, R205 (2007)ADSCrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Znojil, M., Gemperle, F., Mustafa, O.: J. Phys. A: Math. Gen. 35, 5781 (2002)ADSCrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Znojil, M.: Phys. Lett. A 374, 807 (2010)ADSCrossRefMATHGoogle Scholar
  22. 22.
    Znojil, M., Jakubský, V.: J. Phys. A: Math. Gen. 38, 5041 (2005)ADSCrossRefMATHGoogle Scholar
  23. 23.
    Caliceti, E., Graffi, S., Maioli, M.: Commun. Math. Phys. 75, 51 (1980)ADSCrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Alvarez, G.: J. Phys. A: Math. Gen. 27, 4589 (1995)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Nuclear Physics Institute ASCR250 68 ŘežCzech Republic

Personalised recommendations