On the Existence of Bound States of a System of Two Fermions on the Two-Dimensional Cubic Lattice

We construct a two-particle discrete Schrödinger-type operator $$\widehat{H}_{\mu}(k)=\widehat{H}_{0}(k)+\mu\widehat{V}$$ , $$k\in\mathbb{T}^{2}$$ associated to a system of two fermions on the two-dimensional cubic lattice $$\mathbb{Z}^{2}$$ interacting via short-range potential, where the non-perturbed part $$\widehat{H}_{0}(k),\,k\in\mathbb{T}^{2}$$ is a convolution type operator with dispersion relation $$\mathcal{E}_{k}(\cdot)$$ defined on the torus $$\mathbb{T}^{2}$$ and having a degenerate minimum at $$0\in\mathbb{T}^{2}$$ . The existence of eigenvalues below the essential spectrum of the operator $$\widehat{H}_{\mu}(k)$$ is proved in the following two cases: in the case $$k=0$$ for all $$\mu>0$$ and in the case of $$k\neq 0$$ for large $$\mu>0$$ .

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