Abstract
We consider the Schrödinger operator \(H({\mathbf{k}}) = {{H}_{0}}({\mathbf{k}}) - V\) , \({\mathbf{k}} \in {{\mathbb{T}}^{2}},\) associated with a system of two particles on a two-dimensional lattice. It is shown that the subspaces of even as well as odd functions are invariant under operator \(H({\mathbf{k}}).\) The sets of quasimomenta \(\mathcal{K}(1),\) \(\mathcal{K}(2)\) and the class of potentials \({\text{P}}(1),\) \({\text{P}}(2)\) are described, for which the operator \(H({\mathbf{k}})\) has an infinite number of eigenvalues \({{z}_{n}}({\mathbf{k}})\) , \(n \in {{\mathbb{Z}}_{ + }}\) , for \({\mathbf{k}} \in \mathcal{K}(j)\) , \({\hat {v}} \in {\text{P}}(j)\) . The explicit form of \({{z}_{n}}({\mathbf{k}})\) and the rate of convergence of the sequence \({{z}_{n}}({\mathbf{k}})\) to the bottom of the essential spectrum are found.
