Advertisement

International Journal of Theoretical Physics

, Volume 46, Issue 8, pp 1914–1928 | Cite as

Singularities Caused by Coalesced Complex Eigenvalues of an Effective Hamilton Operator

  • I Rotter
  • A. F. Sadreev
Article

Abstract

The S matrix theory with use of the effective Hamiltonian is sketched and applied to the description of the transmission through double quantum dots. The effective Hamilton operator is non-hermitian, its eigenvalues are complex, the eigenfunctions are bi-orthogonal. In this theory, singularities occur at points where two (or more) eigenvalues of the effective Hamiltonian coalesce. These points are physically meaningful: they separate the scenario of avoided level crossings from that without any crossings in the complex plane. They are branch points in the complex plane. Their geometrical features are different from those of the diabolic points.

Keywords

effective Hamilton complex eigenvalue quantum dots branch points 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berry, M. V. (1984). Proc. R. Soc. London, Ser. A 392, 45.MATHADSMathSciNetCrossRefGoogle Scholar
  2. Berry, M. V. and Wilkinson, M. (1984). Proceedings of the Royal Society of London A 392, 15.MATHADSMathSciNetCrossRefGoogle Scholar
  3. Dembowski, C., Dietz, B., Gräf, H. D., Harney, H. L., Heine, A., Heiss, W. D., and Richter, A. (2003). Physical Review Letters 90, 034101.CrossRefADSGoogle Scholar
  4. Dembowski, C., Dietz, B., Gräf, H. D., Harney, H. L., Heine, A., Heiss, W. D., and Richter, A. (2004). Physical Review E 69, 056216.CrossRefADSGoogle Scholar
  5. Dembowski, C., Gräf, H. D., Harney, H. L., Heine, A., Heiss, W. D., Rehfeld, H., and Richter, A. (2001). Physical Reviews Letters 86, 787.CrossRefADSGoogle Scholar
  6. Feshbach, H. (1958). Annals of Physics (New York) 5, 357.MATHCrossRefADSMathSciNetGoogle Scholar
  7. Feshbach, H. (1962). Annals of Physics (New York) 19, 287.MATHCrossRefADSMathSciNetGoogle Scholar
  8. Heiss, W. D. (1999). European Physical Journal D 7, 1.CrossRefADSGoogle Scholar
  9. Heiss, W. D. (2000). Physical Review E 61, 929.CrossRefADSGoogle Scholar
  10. Heiss, W. D., Müller, M., and Rotter, I. (1998). Physical Review E 58, 2894.CrossRefADSGoogle Scholar
  11. Hernández, E., Jáuregui, A., and Mondragon, A. (2000). Journal of Physics A: Mathematical and General 33, 4507.MATHCrossRefADSMathSciNetGoogle Scholar
  12. Hernández, E., Jáuregui, A., and Mondragon, A. (2003). Physical Review A 67, 022721.CrossRefADSGoogle Scholar
  13. In (Heiss et al., 1998; Heiss, 1999, 2000; Dembowski et al., 2001, 2003, 2004; Keck et al., 2003), the singular points are called ‘exceptional points’ although the function space at these points is not incomplete. The incompleteness of the Hilbert space is characteristic of exceptional points (Kato, 1966).Google Scholar
  14. Kato, T. (1966). Perturbation Theory of Linear Operators, Springer, Berlin.Google Scholar
  15. Keck, F., Korsch, H. J., and Mossmann, S. (2003). Journal of Physics A: Mathematical and General 36, 2125.MATHCrossRefADSMathSciNetGoogle Scholar
  16. Kobayashi, K., Aikawa, H., Katsumoto, S., and Iye, Y. (2002). Physical Review Letters 88, 256806.CrossRefADSGoogle Scholar
  17. Ladron de Guevara, M. L., Claro, F., and Orellana, P. A. (2003). Physical Review B 67, 195335.CrossRefADSGoogle Scholar
  18. Lauber, H. M., Weidenhammer, P., and Dubbers, D. (1994). Physical Review Letters 72, 1004.CrossRefADSGoogle Scholar
  19. Magunov, A. I., Rotter, I., and Strakhova, S. I. (1999). Journal of Physics B: Atomic, Molecular and Optical Physics 32, 1669.CrossRefADSGoogle Scholar
  20. Magunov, A. I., Rotter, I., and Strakhova, S. I. (2001). Journal of Physics B: Atomic, Molecular and Optical Physics 34, 29.CrossRefADSGoogle Scholar
  21. Magunov, A. I., Rotter, I., and Strakhova, S. I. (2003a). Journal of Physics B: Atomic, Molecular and Optical Physics 36, L401.CrossRefADSGoogle Scholar
  22. Magunov, A. I., Rotter, I., and Strakhova, S. I. (2003b). Physical Review B 68, 245305.CrossRefADSGoogle Scholar
  23. Moiseyev, N. (1998). Physics Reports 302, 211.CrossRefADSGoogle Scholar
  24. Mondragon, A. and Hernanández, E. (1993). Journal of Physics A: Mathematical and General 26, 5595.CrossRefADSMathSciNetGoogle Scholar
  25. Mondragon, A. and Hernanández, E. (1996). Journal of Physics A: Mathematical and General 29, 2567.MATHCrossRefADSMathSciNetGoogle Scholar
  26. Okołowicz, J., Płoszajczak, M., and Rotter, I. (2003). Physics Reports 374, 271.CrossRefADSMathSciNetMATHGoogle Scholar
  27. Persson, E., Rotter, I., Stöckmann, H. J., and Barth, M. (2000). Physical Review Letters 85, 2478.CrossRefADSGoogle Scholar
  28. Peskin et al., 1994, 1997; Rotter, 1997 Peskin, U., Reisler, H., and Miller, W. H. (1994). Journal of Chemical Physics 101, 9672.CrossRefADSGoogle Scholar
  29. Peskin, U., Reisler, H., and Miller, W. H. (1997). Journal of Chemical Physics 106, 4812.CrossRefADSGoogle Scholar
  30. Rotter, I. (1991). Reports on Progress in Physics 54, 635.CrossRefADSGoogle Scholar
  31. Rotter, I. (1997). Journal of Chemical Physics 106, 4810.CrossRefADSGoogle Scholar
  32. Rotter, I. (2001). Physical Review E 64, 036213.CrossRefADSGoogle Scholar
  33. Rotter, I. (2003). Physical Review E 68, 016211.CrossRefADSMathSciNetGoogle Scholar
  34. Rotter, I. (2004). From discrete states to resonance states: quantum mechanics of systems embedded in a continuum Proceedings of the XXV International Colloquium on Group Theoretical Methods in Physics, Cocoyoc, Mexico, 2–6 August 2004, Edited by: George S. Pogosyan, Luis Edgar Vicent, and Kurt Bernardo Wolf.Google Scholar
  35. Rotter, I. and Sadreev, A. F. (2004). Physical Review E 69, 066201.CrossRefADSGoogle Scholar
  36. Rotter, I. and Sadreev, A. F. (2005a). Physical Review E 71, 036227.Google Scholar
  37. Rotter, I. and Sadreev, A. F. (2005b). Physical Review E 71, 046204.Google Scholar
  38. Sadreev, A. F. and Rotter, I. (2003). Journal of Physics A: Mathematical and General 36, 11413.MATHCrossRefADSMathSciNetGoogle Scholar
  39. Stöckmann, H. J., Persson, E., Kim, Y. H., Barth, M., Kuhl, U., and Rotter, I. (2002). Physical Review E 65, 066211.CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Max-Planck-Institut für Physik komplexer SystemeDresdenGermany
  2. 2.Kirensky Institute of PhysicsKrasnoyarskRussia
  3. 3.Department of Physics and Measurement TechnologyLinköping UniversityLinköpingSweden

Personalised recommendations