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Advances in Mathematical Physics
Volume 2012 (2012), Article ID 309398, 12 pages
http://dx.doi.org/10.1155/2012/309398
Research Article

Resonances for Perturbed Periodic Schrödinger Operator

Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 351, Cours de la Libération, 33405 Talence, France

Received 29 September 2011; Accepted 27 November 2011

Academic Editor: Ali Mostafazadeh

Copyright © 2012 Mouez Dimassi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. V. S. Buslaev, “Semi-classical approximation for equations with periodic coefficients,” Russian Mathematical Surveys, vol. 42, pp. 97–125, 1987. View at Publisher · View at Google Scholar
  2. M. C. Chang and Q. Niu, “Berry phase, hyperorbits, and the Hofstadter spectrum,” Physical Review Letters, vol. 75, pp. 1348–1351, 1996.
  3. M. C. Chang and Q. Niu, “Berry phase, hyperorbits, and the Hofstadter spectrum: semiclassical in magnetic Bloch bands,” Physical Review B, vol. 53, pp. 7010–7022, 1996. View at Publisher · View at Google Scholar
  4. M. Dimassi, J. C. Guillot, and J. Ralston, “Semiclassical asymptotics in magnetic Bloch bands,” Journal of Physics, vol. 35, no. 35, pp. 7597–7605, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. Dimassi, J.-C. Guillot, and J. Ralston, “On effective Hamiltonians for adiabatic perturbations of magnetic Schrödinger operators,” Asymptotic Analysis, vol. 40, no. 2, pp. 137–146, 2004.
  6. C. Gérard, A. Martinez, and J. Sjöstrand, “A mathematical approach to the effective Hamiltonian in perturbed periodic problems,” Communications in Mathematical Physics, vol. 142, no. 2, pp. 217–244, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. F. Hôvermann, H. Spohn, and S. Teufel, “Semiclassical limit for the Schrödinger equation with a short scale periodic potential,” Communications in Mathematical Physics, vol. 215, no. 3, pp. 609–629, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. W. Horn, “Semi-classical constructions in solid state physics,” Communications in Partial Differential Equations, vol. 16, no. 2-3, pp. 255–289, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. G. Nenciu, “Dynamics of band electrons in electric and magnetic fields rigorous justification of the effective Hamiltonians,” Reviews of Modern Physics, vol. 63, no. 1, pp. 91–127, 1991. View at Publisher · View at Google Scholar
  10. R. Peierls, “Zur Theorie des diamagnetimus von leitungselektronen,” Zeitschrift für Physikalische, vol. 80, pp. 763–791, 1933. View at Publisher · View at Google Scholar
  11. J.C. Slater, “Electrons in perturbed periodic lattices,” Physical Review, vol. 76, pp. 1592–1600, 1949. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. N. E. Firsova, “Resonances of Hill operator perturbed by an exponentially decreasing additive potential,” Mathematical Notes, vol. 36, pp. 854–861, 1984.
  13. M. Dimassi, “Resonances for slowly varying perturbations of a periodic Schrödinger operator,” Canadian Journal of Mathematics, vol. 54, no. 5, pp. 998–1037, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. M. Dimassi and M. Zerzeri, “A local trace formula for resonances of perturbed periodic Schrödinger operators,” Journal of Functional Analysis, vol. 198, no. 1, pp. 142–159, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. J. Sjöstrand, “A trace formula for resonances and application to semi-classical Schrödinger operators,” in Séminaire Équations aux Dérivées Partielles, exposé no. 11 (1996-97), 1996.
  16. M. Dimassi and M. Mnif, “Lower bounds for the counting function of resonances for a perturbation of a periodic Schrödinger operator by decreasing potential,” Comptes Rendus Mathématique, vol. 335, no. 12, pp. 1013–1016, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. E. Balslev and J. M. Combes, “Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions,” Communications in Mathematical Physics, vol. 22, pp. 280–294, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. L. Nédeléc, “Resonances for matrix Schrödinger operators,” Duke Mathematical Journal, vol. 106, no. 2, pp. 209–236, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  19. J. Ralston, “Magnetic breakdown,” Astérisque, no. 210, pp. 263–282, 1992. View at Zentralblatt MATH