Perturbation theory for the two-particle Schrodinger operator on a one-dimensional lattice

We consider the two-particle Schrodinger operator H(k) on the one-dimensional lattice Z. The operator H(pi) has infinitely many eigenvalues z(m)(pi) = (v) over cap (m), m is an element of Z(+). If the potential (v) over cap increases on Z(+), then only the eigenvalue z(0)(pi) is simple, and all the other eigenvalues are of multiplicity two. We prove that for each of the doubly degenerate eigenvalues z(m)(pi), m is an element of N, the operator H(pi) splits into two nondegenerate z(m)(-)(k) < z(m)(+)(k) under small variations of k is an element of (pi - delta, pi ). We show that z(m)(-)(k) < z(m)(+)(k) and obtain an estimate for z(m)(+)(k) - z(m)(-)(k) for k is an element of (pi - delta, pi). The eigenvalues z(0)(k) and z(1)(-)(k) increase on [pi - 6, pi]. If (Delta(v) over cap)(m) > 0, then z(m)(+/-)(k) for m >= 2 also has this property.