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Advances in Mathematical Physics
Volume 2014 (2014), Article ID 943868, 10 pages
http://dx.doi.org/10.1155/2014/943868
Research Article

The Faddeev Equation and the Essential Spectrum of a Model Operator Associated with the Hamiltonian of a Nonconserved Number of Particles

1Department of Mathematics, Faculty of Science, University Putra Malaysia, 43400 Serdang, Malaysia
2Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Malaysia
3Academic Lyceum Number One Under Samarkand Institute of Economics and Service, 140104 Samarkand, Uzbekistan

Received 13 February 2014; Accepted 9 April 2014; Published 2 June 2014

Academic Editor: Pavel Kurasov

Copyright © 2014 Zahriddin Muminov et al. 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. R. A. Minlos and H. Spohn, “The three-body problem in radioactive decay: the case of one atom and at most two photons,” in TopicS in StatiStical and Theoretical PhySicS. Berezin’S Memory Volume, American Mathematical Society Translation, vol.177, no. 2, pp. 159193, 1996.
  2. V. A. Malyshev and R. A. Minlos, Linear Infinite-Particle Operators, vol. 143, American Mathematical Society, Providence, RI, USA, 1995. View at MathSciNet
  3. D. C. Mattis, “The few-body problem on a lattice,” Reviews of Modern Physics, vol. 58, no. 2, pp. 361–379, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  4. A. I. Mogilner, “Hamiltonians in solid-state physics as multiparticle discrete Schrödinger operators: problems and results,” in Advances in Soviet Mathematics, vol. 5, pp. 139–194, 1991. View at MathSciNet
  5. K. O. Friedrichs, Perturbation of Spectra in Hilbert Space, American Mathematical Society, Providence, RI, USA, 1965. View at MathSciNet
  6. C. H. Buhler, S. Yunoki, and A. Moreo, “Magnetic domains and stripes in a spin-fermion model for cuprates,” Physical Review Letters, vol. 84, no. 12, Article ID 269026, 2000. View at Publisher · View at Google Scholar
  7. I. M. Sigal, A. Soffer, and L. Zielinski, “On the spectral properties of Hamiltonians without conservation of the particle number,” Journal of Mathematical Physics, vol. 43, no. 4, pp. 1844–1855, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Y. V. Zhukov and R. A. Minlos, “Spectrum and scattering in a “spin-boson” model with not more than three photons,” Theoretical and Mathematical Physics, vol. 103, no. 1, pp. 398–411, 1995. View at Publisher · View at Google Scholar · View at Scopus
  9. S. Albeverio, S. N. Lakaev, and T. H. Rasulov, “On the spectrum of an hamiltonian in fock space. Discrete spectrum asymptotics,” Journal of Statistical Physics, vol. 127, no. 2, pp. 191–220, 2007. View at Publisher · View at Google Scholar · View at Scopus
  10. S. Albeverio, S. N. Lakaev, and T. H. Rasulov, “The Efimov effect for a model operator associated with the Hamiltonian of a non conserved number of particles,” Methods of Functional Analysis and Topology, vol. 13, no. 1, pp. 1–16, 2007. View at Zentralblatt MATH · View at MathSciNet
  11. S. N. Lakaev and T. K. Rasulov, “A model in the theory of perturbations of the essential spectrum of multiparticle operators,” Mathematical Notes, vol. 73, no. 3-4, pp. 521–528, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. T. H. Rasulov, “The Faddeev equation and the location of the essential spectrum of a model operator for several particles,” Mathematics (Izvestiya VUZ. Matematika), vol. 12, pp. 59–69, 2008.
  13. T. H. Rasulov, “On the structure of the essential spectrum of a model many-body Hamiltonian,” Mathematical Notes, vol. 83, no. 1, pp. 86–94, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  14. T. H. Rasulov, “Investigation of the spectrum of a model operator in a Fock space,” Teoreticheskaya i Matematicheskaya Fizika, vol. 161, no. 2, pp. 164–175, 2009 (Russian). View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. T. H. Rasulov, “Study of the essential spectrum of a matrix operator,” Teoreticheskaya i Matematicheskaya Fizika, vol. 164, no. 1, pp. 62–77, 2010 (Russian). View at Publisher · View at Google Scholar
  16. G. R. Yodgarov and M. I. Muminov, “On the spectrum of a model operator in the theory of perturbations of the essential spectrum,” Teoreticheskaya i Matematicheskaya Fizika, vol. 144, no. 3, pp. 544–554, 2005 (Russian). View at Publisher · View at Google Scholar · View at MathSciNet
  17. S.N. Lakaev and M.K. Shermatov, “The spectrum of the Hamiltonian of a system of three quantum particles on a lattice,” Uspekhi Matematicheskikh Nauk, vol. 54, no. 6 (330), pp. 165–166, 1999.
  18. Z. I. Muminov, F. Ismail, Z. K. Eshkuvatov, and J. Rasulov, “On the discrete spectrum of a model operator in fermionic Fock space,” Abstract and Applied Analysis, vol. 2013, Article ID 875194, 12 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  19. Z. I. Muminov, F. Ismail, and Z. K. Eshkuvatov, “On the number of eigenvalues of a model operator in fermionic Fock space,” Journal of Physics: Conference Series, vol. 435, Article ID 012036, 2013. View at Publisher · View at Google Scholar
  20. S. Albeverio, S. N. Lakaev, and Z. I. Muminov, “Schrödinger operators on lattices. The Efimov effect and discrete spectrum asymptotics,” Annales Henri Poincaré, vol. 5, no. 4, pp. 743–772, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. 4 of Analysis of Operators, Academic Press, New York, NY, USA, 1978. View at MathSciNet
  22. M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. 1 of Functional Analysis, Academic Press, New York, NY, USA, 1972. View at MathSciNet
  23. L. D. Faddeev and S. P. Merkuriev, Quantum Scattering Theory for Several Particle Systems, Kluwer Academic, 1993. View at MathSciNet