Examples



mdbootstrap.com



 
Статья
2015

On compact perturbation of two-particle Schrödinger operator on a lattice


M. E. MuminovM. E. Muminov, A. M. KhurramovA. M. Khurramov
Русская математика
https://doi.org/10.3103/S1066369X15060043
Abstract / Full Text

We consider a system of two arbitrary quantum particles on three-dimensional lattice with certain dispersion functions (describing the transfer of the particle from one node to another) interacting by the attractive potential only on the nearest neighboring nodes. We find a class of potentials such that under perturbation of the two-particle operator h(k), corresponding to a two-particle system, with a potential from this class, the discrete spectrum of h(k) is preserved.

Author information
  • Universiti Teknologi Malaysia, Jahor, 81310, Bahru, MalaysiaM. E. Muminov
  • Samarkand State University, Universitetskii bulv. 15, Samarkand, 140101, Republic of UzbekistanA. M. Khurramov
References
  1. Fadeev, L. D. “Mathematical Aspects of the Three-Body Problemin the Quantum Scattering,” Proc. Steklov Institute of Mathematics LXIX (1963).
  2. Mattis, D. C. “The Few-Body Problem on Lattice,” Rew. Mod. Phys. 58, 361–379 (1986).
  3. Albeverio, S., Lakaev, S. N., Makarov, K. A., Muminov, Z. I. “The Threshold Effects for the Two-Particle Hamiltonians,” Commun. Math. Phys. 262, 91–115 (2006).
  4. Yafaev, D. R. “On the Theory of the Discrete Spectrum of the Three-Particle Schrödinger Operator,” Mathematics of the USSR-Sbornik 23, No. 4, 535–559 (1974).
  5. Sobolev, A. V. “The Efimov Effect. Discrete Spectrum. Asymptotics,” Commun. Math. Phys. 156, 101–126 (1993).
  6. Yafaev, D. R. “On the Finiteness of the Discrete Spectrum of the Three-Particle Schrödinger Operator,” Theoretical and Mathematical Physics 25, No. 2, 1065–1072 (1975).
  7. Vugalter, S. A., Zhislin, G. M. “The Spectrum of the Schrödinger Operators of Multiparticle Systems with Short-Range Potentials,” Trudy Mosk.Matem. Obshch. 49, 95–112, (1986).
  8. Zhislin, G. M. “Virtual Levels of n-Particle Systems,” Theoretical and Mathematical Physics 68, No. 2, 815–823 (1986).
  9. Lakaev, S. N., Tilavova, Sh. M. “Merging of Eigenvalues and Resonances of a Two-Particle Schrödinger Operator,” Theoretical and Mathematical Physics 101, No. 2, 1320–1331 (1994).
  10. Abdullaev, Zh. I., Lakaev, S. N. “Finiteness of Discrete Spectrum of Three-Particle Schrödinger Operator on a lattice,” Theoretical and Mathematical Physics 111, No. 1, 467–479 (1997).
  11. Lakaev, S. N., Bozorov, I. N. “The Number of Bound States of a One-Particle Hamiltonian on a Three-Dimensional Lattice,” Theoretical and Mathematical Physics 158, No. 3, 360–376 (2009).
  12. Muminov, M. E., Khurramov, A. M. “Spectral Properties of a Two-Particle Hamiltonian on a Lattice,” Theoretical and Mathematical Physics 177, No. 3, 1693–1705 (2013).
  13. Reed, M., Simon, B. Methods of Modern Mathematical Physics. Analysis of Operators (Academic Press, New York-San Francisco-London, 1978; Mir, Moscow, 1982), Vol. 4.