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Theoretical and Mathematical Physics

, Volume 145, Issue 2, pp 1551–1558 | Cite as

Perturbation Theory for the Two-Particle Schrodinger Operator on a One-Dimensional Lattice

  • J. I. Abdullaev
Article

Abstract

We consider the two-particle Schrodinger operator H(k) on the one-dimensional lattice ℤ. The operator H(π) has infinitely many eigenvalues zm(π) = v(m), m ∈ ℤ+. If the potential v increases on ℤ+, then only the eigenvalue z0(π) is simple, and all the other eigenvalues are of multiplicity two. We prove that for each of the doubly degenerate eigenvalues zm(π), m ∈ ℕ, the operator H(π) splits into two nondegenerate eigenvalues z m (k) and z m + (k) under small variations of k ∈ (π − δ, π). We show that z m (k) < z m + (k) and obtain an estimate for z m + (k) − z m (k) for k ∈ (π − δ, π). The eigenvalues z0(k) and z 1 (k) increase on [π − δ, π]. If (Δv)(m) > 0, then z m ± (k) for m ≥ 2 also has this property.

Keywords

Hamiltonian Schrodinger operator total quasimomentum eigenvalue perturbation theory 

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REFERENCES

  1. 1.
    T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1966).Google Scholar
  2. 2.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators, Acad. Press, New York (1978).Google Scholar
  3. 3.
    S. N. Lakaev, Theor. Math. Phys., 44, 810–814 (1980); J. Rauch, J. Funct. Anal., 35, 304–315 (1980); B. Simon, Ann. Phys., 97, 279–288 (1976).CrossRefMathSciNetGoogle Scholar
  4. 4.
    Sh. S. Mamatov and R. A. Minlos, Theor. Math. Phys., 79, 455–466 (1989).CrossRefMathSciNetGoogle Scholar
  5. 5.
    R. A. Minlos and A. I. Mogilner, “Some problems concerning spectra of lattice models,” in: Schodinger Operators, Standard and Nonstandard (Proc. Conf. in Dubna, USSR, 6–10 September 1988, P. Exner and P. Seba, eds.), World Scientific, Singapore (1989), pp. 243–257.Google Scholar
  6. 6.
    Zh. I. Abdullaev, I. A. Ikromov, and S. N. Lakaev, Theor. Math. Phys., 103, 390–397 (1995).Google Scholar
  7. 7.
    P. A. Faria da Viega, L. Ioriatti, and M. O'Carrol, Phys. Rev. E, 66, 016130 (2002); D. C. Mattis, Rev. Modern Phys., 58, 361–379 (1986).ADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • J. I. Abdullaev
    • 1
  1. 1.Samarkand State UniversitySamarkandUzbekistan

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