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


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.


Hamiltonian Schrodinger operator total quasimomentum eigenvalue perturbation theory 


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© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

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

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