On the Infinite Number of Eigenvalues of the Two-Particle Schrödinger Operator on a Lattice

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.

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