A Two-Boson 2D Lattice Hamiltonian with Interactions up to Next-to-Neighboring Sites
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.