Abstract
We consider the lattice Schrödinger operator \(H_{\gamma\lambda\mu}(K)\) associated with a system of two identical spinless bosons on the two-dimensional square lattice \(\mathbb{Z}^{2}\) . It is assumed that the center-of-mass quasimomentum \(K\) equals zero and that the bosons only interact with each other on-site and on the first and second nearest neighboring sites in the lattice. These interactions have magnitudes \(\gamma\) , \(\lambda\) and \(\mu\) , respectively. We prove the existence of an invariant subspace for \(H_{\gamma\lambda\mu}(0)\) such that its restriction, \(H^{\rm ees}_{\gamma\lambda\mu}(0)\) , has at most four eigenvalues. In addition, we partition the \((\gamma,\lambda,\mu)\) -space into connected components such that, in each component, the operator \(H^{\textrm{ees}}_{\gamma\lambda\mu}(0)\) has fixed numbers of eigenvalues below the bottom of the essential spectrum and above its top.
