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

Abstract

We consider the two-particle Schrodinger operator H(k) on the one-dimensional lattice ℤ. The operator H(π) has infinitely many eigenvalues z_m(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 z_m(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.