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 $\mathbb Z$. The operator $H(\pi)$ has infinitely many eigenvalues $z_m(\pi)=\hat v(m)$, $m\in\mathbb Z_+$. If the potential $\hat v$ increases on $\mathbb Z_+$, 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 $z_m(\pi)$, $m\in\mathbb N$, the operator $H(\pi)$ splits into two nondegenerate eigenvalues $z_m^-(k)$ and $z_m^+(k)$ under small variations of $k\in(\pi-\delta,\pi)$. We show that $z_m^-(k)<z_m^+(k)$ and obtain an estimate for $z_m^+(k)-z_m^-(k)$ for при $k\in(\pi-\delta,\pi)$. The eigenvalues $z_0(k)$ and $z_1^-(k)$ increase on$[\pi-\delta,\pi]$. If $(\Delta\hat v)(m)>0$, then $z_m^\pm(k)$ for $m\geqslant 2$ also has this property.