Existence and analyticity of bound states of a two-particle Schrödinger operator on a lattice

Abstract

We consider the two-particle discrete Schrodinger operator Hμ(K) corresponding to a system of two arbitrary particles on a d-dimensional lattice ℤd, d ≥ 3, interacting via a pair contact repulsive potential with a coupling constant μ > 0 (\(K \in \mathbb{T}^d\) is the quasimomentum of two particles). We find that the upper (right) edge of the essential spectrum can be either a virtual level (for d = 3, 4) or an eigenvalue (for d ≥ 5) of Hμ (K). We show that there exists a unique eigenvalue located to the right of the essential spectrum, depending on the coupling constant μ and the two-particle quasimomentum K. We prove the analyticity of the corresponding eigenstate and the analyticity of the eigenvalue and the eigenstate as functions of the quasimomentum \(K \in \mathbb{T}^d\) in the domain of their existence.