Asymptotic behavior of eigenvalues of the two-particle discrete Schrödinger operator

Abstract

We consider two-particle Schrodinger operator H(k) on a three-dimensional lattice ℤ 3 (here k is the total quasimomentum of a two-particle system, $$k \in \mathbb{T}^3 : = \left( { - \pi ,\pi ]^3 } \right)$$ . We show that for any $$k \in S = \mathbb{T}^3 \backslash ( - \pi ,\pi )^3$$ , there is a potential $$\hat v$$ such that the two-particle operator H(k) has infinitely many eigenvalues zn(k) accumulating near the left boundary m(k) of the continuous spectrum. We describe classes of potentials W(j) and W(ij) and manifolds S(j) ⊂ S, i, j ∈ {1, 2, 3}, such that if k ∈ S(3), (k 2 , k 3 ) ∈ (−π,π) 2 , and $$\hat v \in W(3)$$ , then the operator H(k) has infinitely many eigenvalues zn(k) with an asymptotic exponential form as n → ∞ and if k ∈ S(i) ∩ S(j) and $$\hat v \in W(ij)$$ , then the eigenvalues znm(k) of H(k) can be calculated exactly. In both cases, we present the explicit form of the eigenfunctions.